Metrizable topological vector space

 ( Metric Spaces ) 

In functional analysis and related areas of mathematics, a metrizable (resp. pseudometrizable) topological vector space (TVS) is a TVS whose topology is induced by a metric (resp. pseudometric). An LM-space is an inductive limit of a sequence of locally convex metrizable TVS.

---

Pseudometrics and metrics A pseudometric on a set $X$ is a map $d\; :\; X\; \backslash times\; X\; \backslash rarr\; \backslash R$ satisfying the following properties:

1. $d(x,\; x)\; =\; 0\; \backslash text\{\; for\; all\; \}\; x\; \backslash in\; X$;
2. Symmetry: $d(x,\; y)\; =\; d(y,\; x)\; \backslash text\{\; for\; all\; \}\; x,\; y\; \backslash in\; X$;
3. Subadditive: $d(x,\; z)\; \backslash leq\; d(x,\; y)\; +\; d(y,\; z)\; \backslash text\{\; for\; all\; \}\; x,\; y,\; z\; \backslash in\; X.$

A pseudometric is called a metric if it satisfies:

4. Identity of indiscernibles: for all $x,\; y\; \backslash in\; X,$ if $d(x,\; y)\; =\; 0$ then $x\; =\; y.$

Ultrapseudometric

A pseudometric $d$ on $X$ is called a Ultrametric or a strong pseudometric if it satisfies:

5. Strong/ Ultrametric triangle inequality: $d(x,\; z)\; \backslash leq\; \backslash max\; \backslash \{\; d(x,\; y),\; d(y,\; z)\; \backslash \}\; \backslash text\{\; for\; all\; \}\; x,\; y,\; z\; \backslash in\; X.$

Pseudometric space

A pseudometric space is a pair $(X,\; d)$ consisting of a set $X$ and a pseudometric $d$ on $X$ such that $X$'s topology is identical to the topology on $X$ induced by $d.$ We call a pseudometric space $(X,\; d)$ a metric space (resp. ultrapseudometric space) when $d$ is a metric (resp. ultrapseudometric).

---

Topology induced by a pseudometric If $d$ is a pseudometric on a set $X$ then collection of open balls: as $z$ ranges over $X$ and $r\; >\; 0$ ranges over the positive real numbers, forms a basis for a topology on $X$ that is called the $d$ -topology or the pseudometric topology on $X$ induced by $d.$

: If $(X,\; d)$ is a pseudometric space and $X$ is treated as a topological space, then unless indicated otherwise, it should be assumed that $X$ is endowed with the topology induced by $d.$

Pseudometrizable space

A topological space $(X,\; \backslash tau)$ is called pseudometrizable (resp. metrizable, ultrapseudometrizable) if there exists a pseudometric (resp. metric, ultrapseudometric) $d$ on $X$ such that $\backslash tau$ is equal to the topology induced by $d.$

---

Pseudometrics and values on topological groups An additive topological group is an additive group endowed with a topology, called a group topology, under which addition and negation become continuous operators.

A topology $\backslash tau$ on a real or complex vector space $X$ is called a vector topology or a TVS topology if it makes the operations of vector addition and scalar multiplication continuous (that is, if it makes $X$ into a topological vector space).

Every topological vector space (TVS) $X$ is an additive commutative topological group but not all group topologies on $X$ are vector topologies. This is because despite it making addition and negation continuous, a group topology on a vector space $X$ may fail to make scalar multiplication continuous. For instance, the discrete topology on any non-trivial vector space makes addition and negation continuous but do not make scalar multiplication continuous.

---

Translation invariant pseudometrics If $X$ is an additive group then we say that a pseudometric $d$ on $X$ is translation invariant or just invariant if it satisfies any of the following equivalent conditions:

1. Translation invariance: $d(x\; +\; z,\; y\; +\; z)\; =\; d(x,\; y)\; \backslash text\{\; for\; all\; \}\; x,\; y,\; z\; \backslash in\; X$;
2. $d(x,\; y)\; =\; d(x\; -\; y,\; 0)\; \backslash text\{\; for\; all\; \}\; x,\; y\; \backslash in\; X.$

---

Value/G-seminorm If $X$ is a topological group the a value or G-seminorm on $X$ (the G stands for Group) is a real-valued map $p\; :\; X\; \backslash rarr\; \backslash R$ with the following properties:

1. Non-negative: $p\; \backslash geq\; 0.$
2. Subadditive: $p(x\; +\; y)\; \backslash leq\; p(x)\; +\; p(y)\; \backslash text\{\; for\; all\; \}\; x,\; y\; \backslash in\; X$;
3. $p(0)\; =\; 0..$
4. Symmetric: $p(-x)\; =\; p(x)\; \backslash text\{\; for\; all\; \}\; x\; \backslash in\; X.$

where we call a G-seminorm a G-norm if it satisfies the additional condition:

5. Total/ Positive definite: If $p(x)\; =\; 0$ then $x\; =\; 0.$

---

Properties of values If $p$ is a value on a vector space $X$ then:

• $|p(x)\; -\; p(y)|\; \backslash leq\; p(x\; -\; y)\; \backslash text\{\; for\; all\; \}\; x,\; y\; \backslash in\; X.$
• $p(n\; x)\; \backslash leq\; n\; p(x)$ and $\backslash frac\{1\}\{n\}\; p(x)\; \backslash leq\; p(x\; /\; n)$ for all $x\; \backslash in\; X$ and positive integers $n.$
• The set $\backslash \{\; x\; \backslash in\; X\; :\; p(x)\; =\; 0\; \backslash \}$ is an additive subgroup of $X.$

---

Equivalence on topological groups

---

Pseudometrizable topological groups

---

An invariant pseudometric that doesn't induce a vector topology Let $X$ be a non-trivial (i.e. $X\; \backslash neq\; \backslash \{\; 0\; \backslash \}$) real or complex vector space and let $d$ be the translation-invariant trivial metric on $X$ defined by $d(x,\; x)\; =\; 0$ and $d(x,\; y)\; =\; 1\; \backslash text\{\; for\; all\; \}\; x,\; y\; \backslash in\; X$ such that $x\; \backslash neq\; y.$ The topology $\backslash tau$ that $d$ induces on $X$ is the discrete topology, which makes $(X,\; \backslash tau)$ into a commutative topological group under addition but does form a vector topology on $X$ because $(X,\; \backslash tau)$ is Connected space but every vector topology is connected. What fails is that scalar multiplication isn't continuous on $(X,\; \backslash tau).$

This example shows that a translation-invariant (pseudo)metric is enough to guarantee a vector topology, which leads us to define paranorms and F-seminorms.

---

Additive sequences A collection $\backslash mathcal\{N\}$ of subsets of a vector space is called additive if for every $N\; \backslash in\; \backslash mathcal\{N\},$ there exists some $U\; \backslash in\; \backslash mathcal\{N\}$ such that $U\; +\; U\; \backslash subseteq\; N.$

All of the above conditions are consequently a necessary for a topology to form a vector topology. Additive sequences of sets have the particularly nice property that they define non-negative continuous real-valued subadditive functions. These functions can then be used to prove many of the basic properties of topological vector spaces and also show that a Hausdorff TVS with a countable basis of neighborhoods is metrizable. The following theorem is true more generally for commutative additive topological groups.

Assume that $n\_\{\backslash bull\}\; =\; \backslash left(n\_1,\; \backslash ldots,\; n\_k\backslash right)$ always denotes a finite sequence of non-negative integers and use the notation: $$\backslash sum\; 2^\{-\; n\_\{\backslash bull\}\}\; :=\; 2^\{-\; n\_1\}\; +\; \backslash cdots\; +\; 2^\{-\; n\_k\}\; \backslash quad\; \backslash text\{\; and\; \}\; \backslash quad\; \backslash sum\; U\_\{n\_\{\backslash bull\}\}\; :=\; U\_\{n\_1\}\; +\; \backslash cdots\; +\; U\_\{n\_k\}.$$

For any integers $n\; \backslash geq\; 0$ and $d\; >\; 2,$ $$U\_n\; \backslash supseteq\; U\_\{n+1\}\; +\; U\_\{n+1\}\; \backslash supseteq\; U\_\{n+1\}\; +\; U\_\{n+2\}\; +\; U\_\{n+2\}\; \backslash supseteq\; U\_\{n+1\}\; +\; U\_\{n+2\}\; +\; \backslash cdots\; +\; U\_\{n+d\}\; +\; U\_\{n+d+1\}\; +\; U\_\{n+d+1\}.$$

From this it follows that if $n\_\{\backslash bull\}\; =\; \backslash left(n\_1,\; \backslash ldots,\; n\_k\backslash right)$ consists of distinct positive integers then $\backslash sum\; U\_\{n\_\{\backslash bull\}\}\; \backslash subseteq\; U\_\{-1\; +\; \backslash min\; \backslash left(n\_\{\backslash bull\}\backslash right)\}.$

It will now be shown by induction on $k$ that if $n\_\{\backslash bull\}\; =\; \backslash left(n\_1,\; \backslash ldots,\; n\_k\backslash right)$ consists of non-negative integers such that $\backslash sum\; 2^\{-\; n\_\{\backslash bull\}\}\; \backslash leq\; 2^\{-\; M\}$ for some integer $M\; \backslash geq\; 0$ then $\backslash sum\; U\_\{n\_\{\backslash bull\}\}\; \backslash subseteq\; U\_M.$ This is clearly true for $k\; =\; 1$ and $k\; =\; 2$ so assume that $k\; >\; 2,$ which implies that all $n\_i$ are positive. If all $n\_i$ are distinct then this step is done, and otherwise pick distinct indices such that $n\_i\; =\; n\_j$ and construct $m\_\{\backslash bull\}\; =\; \backslash left(m\_1,\; \backslash ldots,\; m\_\{k-1\}\backslash right)$ from $n\_\{\backslash bull\}$ by replacing each $n\_i$ with $n\_i\; -\; 1$ and deleting the $j^\{\backslash text\{th\}\}$ element of $n\_\{\backslash bull\}$ (all other elements of $n\_\{\backslash bull\}$ are transferred to $m\_\{\backslash bull\}$ unchanged). Observe that $\backslash sum\; 2^\{-\; n\_\{\backslash bull\}\}\; =\; \backslash sum\; 2^\{-\; m\_\{\backslash bull\}\}$ and $\backslash sum\; U\_\{n\_\{\backslash bull\}\}\; \backslash subseteq\; \backslash sum\; U\_\{m\_\{\backslash bull\}\}$ (because $U\_\{n\_i\}\; +\; U\_\{n\_j\}\; \backslash subseteq\; U\_\{n\_i\; -\; 1\}$) so by appealing to the inductive hypothesis we conclude that $\backslash sum\; U\_\{n\_\{\backslash bull\}\}\; \backslash subseteq\; \backslash sum\; U\_\{m\_\{\backslash bull\}\}\; \backslash subseteq\; U\_M,$ as desired.

It is clear that $f(0)\; =\; 0$ and that $0\; \backslash leq\; f\; \backslash leq\; 1$ so to prove that $f$ is subadditive, it suffices to prove that $f(x\; +\; y)\; \backslash leq\; f(x)\; +\; f(y)$ when $x,\; y\; \backslash in\; X$ are such that which implies that $x,\; y\; \backslash in\; U\_0.$ This is an exercise. If all $U\_i$ are symmetric then $x\; \backslash in\; \backslash sum\; U\_\{n\_\{\backslash bull\}\}$ if and only if $-\; x\; \backslash in\; \backslash sum\; U\_\{n\_\{\backslash bull\}\}$ from which it follows that $f(-x)\; \backslash leq\; f(x)$ and $f(-x)\; \backslash geq\; f(x).$ If all $U\_i$ are balanced then the inequality $f(s\; x)\; \backslash leq\; f(x)$ for all unit scalars $s$ such that $|s|\; \backslash leq\; 1$ is proved similarly. Because $f$ is a nonnegative subadditive function satisfying $f(0)\; =\; 0,$ as described in the article on sublinear functionals, $f$ is uniformly continuous on $X$ if and only if $f$ is continuous at the origin. If all $U\_i$ are neighborhoods of the origin then for any real $r\; >\; 0,$ pick an integer $M\; >\; 1$ such that so that $x\; \backslash in\; U\_M$ implies If the set of all $U\_i$ form basis of balanced neighborhoods of the origin then it may be shown that for any $n\; >\; 1,$ there exists some such that implies $x\; \backslash in\; U\_n.$ $\backslash blacksquare$

---

Paranorms If $X$ is a vector space over the real or complex numbers then a paranorm on $X$ is a G-seminorm (defined above) $p\; :\; X\; \backslash rarr\; \backslash R$ on $X$ that satisfies any of the following additional conditions, each of which begins with "for all sequences $x\_\{\backslash bull\}\; =\; \backslash left(x\_i\backslash right)\_\{i=1\}^\{\backslash infty\}$ in $X$ and all convergent sequences of scalars $s\_\{\backslash bull\}\; =\; \backslash left(s\_i\backslash right)\_\{i=1\}^\{\backslash infty\}$":

1. Continuity of multiplication: if $s$ is a scalar and $x\; \backslash in\; X$ are such that $p\backslash left(x\_i\; -\; x\backslash right)\; \backslash to\; 0$ and $s\_\{\backslash bull\}\; \backslash to\; s,$ then $p\backslash left(s\_i\; x\_i\; -\; s\; x\backslash right)\; \backslash to\; 0.$
2. Both of the conditions:
   • if $s\_\{\backslash bull\}\; \backslash to\; 0$ and if $x\; \backslash in\; X$ is such that $p\backslash left(x\_i\; -\; x\backslash right)\; \backslash to\; 0$ then $p\backslash left(s\_i\; x\_i\backslash right)\; \backslash to\; 0$;
   • if $p\backslash left(x\_\{\backslash bull\}\backslash right)\; \backslash to\; 0$ then $p\backslash left(s\; x\_i\backslash right)\; \backslash to\; 0$ for every scalar $s.$
3. Both of the conditions:
   • if $p\backslash left(x\_\{\backslash bull\}\backslash right)\; \backslash to\; 0$ and $s\_\{\backslash bull\}\; \backslash to\; s$ for some scalar $s$ then $p\backslash left(s\_i\; x\_i\backslash right)\; \backslash to\; 0$;
   • if $s\_\{\backslash bull\}\; \backslash to\; 0$ then $p\backslash left(s\_i\; x\backslash right)\; \backslash to\; 0\; \backslash text\{\; for\; all\; \}\; x\; \backslash in\; X.$
4. Separate continuity:
   • if $s\_\{\backslash bull\}\; \backslash to\; s$ for some scalar $s$ then $p\backslash left(s\; x\_i\; -\; s\; x\backslash right)\; \backslash to\; 0$ for every $x\; \backslash in\; X$;
   • if $s$ is a scalar, $x\; \backslash in\; X,$ and $p\backslash left(x\_i\; -\; x\backslash right)\; \backslash to\; 0$ then $p\backslash left(s\; x\_i\; -\; s\; x\backslash right)\; \backslash to\; 0$ .

A paranorm is called total if in addition it satisfies:

• Total/ Positive definite: $p(x)\; =\; 0$ implies $x\; =\; 0.$

---

Properties of paranorms If $p$ is a paranorm on a vector space $X$ then the map $d\; :\; X\; \backslash times\; X\; \backslash rarr\; \backslash R$ defined by $d(x,\; y)\; :=\; p(x\; -\; y)$ is a translation-invariant pseudometric on $X$ that defines a on $X.$

If $p$ is a paranorm on a vector space $X$ then:

• the set $\backslash \{\; x\; \backslash in\; X\; :\; p(x)\; =\; 0\; \backslash \}$ is a vector subspace of $X.$
• $p(x\; +\; n)\; =\; p(x)\; \backslash text\{\; for\; all\; \}\; x,\; n\; \backslash in\; X$ with $p(n)\; =\; 0.$
• If a paranorm $p$ satisfies $p(s\; x)\; \backslash leq\; |s|\; p(x)\; \backslash text\{\; for\; all\; \}\; x\; \backslash in\; X$ and scalars $s,$ then $p$ is absolutely homogeneity (i.e. equality holds) and thus $p$ is a seminorm.

---

Examples of paranorms

• If $d$ is a translation-invariant pseudometric on a vector space $X$ that induces a vector topology $\backslash tau$ on $X$ (i.e. $(X,\; \backslash tau)$ is a TVS) then the map $p(x)\; :=\; d(x\; -\; y,\; 0)$ defines a continuous paranorm on $(X,\; \backslash tau)$; moreover, the topology that this paranorm $p$ defines in $X$ is $\backslash tau.$
• If $p$ is a paranorm on $X$ then so is the map $q(x)\; :=\; p(x)\; /\; 1.$
• Every positive scalar multiple of a paranorm (resp. total paranorm) is again such a paranorm (resp. total paranorm).
• Every seminorm is a paranorm.
• The restriction of an paranorm (resp. total paranorm) to a vector subspace is an paranorm (resp. total paranorm).
• The sum of two paranorms is a paranorm.
• If $p$ and $q$ are paranorms on $X$ then so is $(p\; \backslash wedge\; q)(x)\; :=\; \backslash inf\_\{\}\; \backslash \{\; p(y)\; +\; q(z)\; :\; x\; =\; y\; +\; z\; \backslash text\{\; with\; \}\; y,\; z\; \backslash in\; X\; \backslash \}.$ Moreover, $(p\; \backslash wedge\; q)\; \backslash leq\; p$ and $(p\; \backslash wedge\; q)\; \backslash leq\; q.$ This makes the set of paranorms on $X$ into a conditionally complete lattice.
• Each of the following real-valued maps are paranorms on $X\; :=\; \backslash R^2$:
  • $(x,\; y)\; \backslash mapsto\; |x|$
  • $(x,\; y)\; \backslash mapsto\; |x|\; +\; |y|$
• The real-valued maps $(x,\; y)\; \backslash mapsto\; \backslash sqrt\{\backslash left|x^2\; -\; y^2\backslash right|\}$ and $(x,\; y)\; \backslash mapsto\; \backslash left|x^2\; -\; y^2\backslash right|^\{3/2\}$ are paranorms on $X\; :=\; \backslash R^2.$
• If $x\_\{\backslash bull\}\; =\; \backslash left(x\_i\backslash right)\_\{i\; \backslash in\; I\}$ is a Hamel basis on a vector space $X$ then the real-valued map that sends $x\; =\; \backslash sum\_\{i\; \backslash in\; I\}\; s\_i\; x\_i\; \backslash in\; X$ (where all but finitely many of the scalars $s\_i$ are 0) to $\backslash sum\_\{i\; \backslash in\; I\}\; \backslash sqrt\{\backslash left|s\_i\backslash right|\}$ is a paranorm on $X,$ which satisfies $p(sx)\; =\; \backslash sqrt$ p(x) for all $x\; \backslash in\; X$ and scalars $s.$
• The function $p(x)\; :=\; |\backslash sin\; (\backslash pi\; x)|\; +\; \backslash min\; \backslash \{\; 2,\; |x|\; \backslash \}$ is a paranorm on $\backslash R$ that is balanced but nevertheless equivalent to the usual norm on $R.$ Note that the function $x\; \backslash mapsto\; |\backslash sin\; (\backslash pi\; x)|$ is subadditive.
• Let $X\_\{\backslash Complex\}$ be a complex vector space and let $X\_\{\backslash R\}$ denote $X\_\{\backslash Complex\}$ considered as a vector space over $\backslash R.$ Any paranorm on $X\_\{\backslash Complex\}$ is also a paranorm on $X\_\{\backslash R\}.$

---

F-seminorms If $X$ is a vector space over the real or complex numbers then an F-seminorm on $X$ (the $F$ stands for Fréchet) is a real-valued map $p\; :\; X\; \backslash to\; \backslash Reals$ with the following four properties:

1. Non-negative: $p\; \backslash geq\; 0.$
2. Subadditive: $p(x\; +\; y)\; \backslash leq\; p(x)\; +\; p(y)$ for all $x,\; y\; \backslash in\; X$
3. Balanced: $p(a\; x)\; \backslash leq\; p(x)$ for $x\; \backslash in\; X$ all scalars $a$ satisfying $|a|\; \backslash leq\; 1;$
   • This condition guarantees that each set of the form $\backslash \{z\; \backslash in\; X\; :\; p(z)\; \backslash leq\; r\backslash \}$ or for some $r\; \backslash geq\; 0$ is a balanced set.
4. For every $x\; \backslash in\; X,$ $p\backslash left(\backslash tfrac\{1\}\{n\}\; x\backslash right)\; \backslash to\; 0$ as $n\; \backslash to\; \backslash infty$
   • The sequence $\backslash left(\backslash tfrac\{1\}\{n\}\backslash right)\_\{n=1\}^\backslash infty$ can be replaced by any positive sequence converging to the zero.

An F-seminorm is called an F-norm if in addition it satisfies:

5. Total/ Positive definite: $p(x)\; =\; 0$ implies $x\; =\; 0.$

An F-seminorm is called monotone if it satisfies:

6. Monotone: for all non-zero $x\; \backslash in\; X$ and all real $s$ and $t$ such that

---

F-seminormed spaces An F -seminormed space (resp. F-normed space) is a pair $(X,\; p)$ consisting of a vector space $X$ and an F-seminorm (resp. F-norm) $p$ on $X.$

If $(X,\; p)$ and $(Z,\; q)$ are F-seminormed spaces then a map $f\; :\; X\; \backslash to\; Z$ is called an isometric embedding if $q(f(x)\; -\; f(y))\; =\; p(x,\; y)\; \backslash text\{\; for\; all\; \}\; x,\; y\; \backslash in\; X.$

Every isometric embedding of one F-seminormed space into another is a topological embedding, but the converse is not true in general.

---

Examples of F-seminorms

• Every positive scalar multiple of an F-seminorm (resp. F-norm, seminorm) is again an F-seminorm (resp. F-norm, seminorm).
• The sum of finitely many F-seminorms (resp. F-norms) is an F-seminorm (resp. F-norm).
• If $p$ and $q$ are F-seminorms on $X$ then so is their pointwise supremum $x\; \backslash mapsto\; \backslash sup\; \backslash \{p(x),\; q(x)\backslash \}.$ The same is true of the supremum of any non-empty finite family of F-seminorms on $X.$
• The restriction of an F-seminorm (resp. F-norm) to a vector subspace is an F-seminorm (resp. F-norm).
• A non-negative real-valued function on $X$ is a seminorm if and only if it is a Convex function F-seminorm, or equivalently, if and only if it is a convex balanced G-seminorm. In particular, every seminorm is an F-seminorm.
• For any the map $f$ on $\backslash Reals^n$ defined by $$f\backslash left(x\_1,^p\; =\; \backslash left|x\_1\backslash right|^p\; +\; \backslash cdots\; \backslash left|x\_n\backslash right|^p$$ is an F-norm that is not a norm.
• If $L\; :\; X\; \backslash to\; Y$ is a linear map and if $q$ is an F-seminorm on $Y,$ then $q\; \backslash circ\; L$ is an F-seminorm on $X.$
• Let $X\_\backslash Complex$ be a complex vector space and let $X\_\backslash Reals$ denote $X\_\backslash Complex$ considered as a vector space over $\backslash Reals.$ Any F-seminorm on $X\_\backslash Complex$ is also an F-seminorm on $X\_\backslash Reals.$

---

Properties of F-seminorms Every F-seminorm is a paranorm and every paranorm is equivalent to some F-seminorm. Every F-seminorm on a vector space $X$ is a value on $X.$ In particular, $p(x)\; =\; 0,$ and $p(x)\; =\; p(-x)$ for all $x\; \backslash in\; X.$

---

Topology induced by a single F-seminorm

---

Topology induced by a family of F-seminorms Suppose that $\backslash mathcal\{L\}$ is a non-empty collection of F-seminorms on a vector space $X$ and for any finite subset $\backslash mathcal\{F\}\; \backslash subseteq\; \backslash mathcal\{L\}$ and any $r\; >\; 0,$ let

The set $\backslash left\backslash \{U\_\{\backslash mathcal\{F\},\; r\}\; ~:~\; r\; >\; 0,\; \backslash mathcal\{F\}\; \backslash subseteq\; \backslash mathcal\{L\},\; \backslash mathcal\{F\}\; \backslash text\{\; finite\; \}\backslash right\backslash \}$ forms a filter base on $X$ that also forms a neighborhood basis at the origin for a vector topology on $X$ denoted by $\backslash tau\_\{\backslash mathcal\{L\}\}.$ Each $U\_\{\backslash mathcal\{F\},\; r\}$ is a Balanced set and Absorbing set subset of $X.$ These sets satisfy $$U\_\{\backslash mathcal\{F\},\; r/2\}\; +\; U\_\{\backslash mathcal\{F\},\; r/2\}\; \backslash subseteq\; U\_\{\backslash mathcal\{F\},\; r\}.$$

• $\backslash tau\_\{\backslash mathcal\{L\}\}$ is the coarsest vector topology on $X$ making each $p\; \backslash in\; \backslash mathcal\{L\}$ continuous.
• $\backslash tau\_\{\backslash mathcal\{L\}\}$ is Hausdorff if and only if for every non-zero $x\; \backslash in\; X,$ there exists some $p\; \backslash in\; \backslash mathcal\{L\}$ such that $p(x)\; >\; 0.$
• If $\backslash mathcal\{F\}$ is the set of all continuous F-seminorms on $\backslash left(X,\; \backslash tau\_\{\backslash mathcal\{L\}\}\backslash right)$ then $\backslash tau\_\{\backslash mathcal\{L\}\}\; =\; \backslash tau\_\{\backslash mathcal\{F\}\}.$
• If $\backslash mathcal\{F\}$ is the set of all pointwise suprema of non-empty finite subsets of $\backslash mathcal\{F\}$ of $\backslash mathcal\{L\}$ then $\backslash mathcal\{F\}$ is a directed family of F-seminorms and $\backslash tau\_\{\backslash mathcal\{L\}\}\; =\; \backslash tau\_\{\backslash mathcal\{F\}\}.$

---

Fréchet combination Suppose that $p\_\{\backslash bull\}\; =\; \backslash left(p\_i\backslash right)\_\{i=1\}^\{\backslash infty\}$ is a family of non-negative subadditive functions on a vector space $X.$

The Fréchet combination of $p\_\{\backslash bull\}$ is defined to be the real-valued map $$p(x)\; :=\; \backslash sum\_\{i=1\}^\{\backslash infty\}\; \backslash frac\{p\_i(x)\}\{2^\{i\}\; \backslash left\}.$$

---

As an F-seminorm Assume that $p\_\{\backslash bull\}\; =\; \backslash left(p\_i\backslash right)\_\{i=1\}^\{\backslash infty\}$ is an increasing sequence of seminorms on $X$ and let $p$ be the Fréchet combination of $p\_\{\backslash bull\}.$ Then $p$ is an F-seminorm on $X$ that induces the same locally convex topology as the family $p\_\{\backslash bull\}$ of seminorms.

Since $p\_\{\backslash bull\}\; =\; \backslash left(p\_i\backslash right)\_\{i=1\}^\{\backslash infty\}$ is increasing, a basis of open neighborhoods of the origin consists of all sets of the form as $i$ ranges over all positive integers and $r\; >\; 0$ ranges over all positive real numbers.

The translation invariant pseudometric on $X$ induced by this F-seminorm $p$ is $$d(x,\; y)\; =\; \backslash sum^\{\backslash infty\}\_\{i=1\}\; \backslash frac\{1\}\{2^i\}\; \backslash frac\{p\_i(\; x\; -\; y\; )\}\{1\; +\; p\_i(\; x\; -\; y\; )\}.$$

This metric was discovered by Fréchet in his 1906 thesis for the spaces of real and complex sequences with pointwise operations.

---

As a paranorm If each $p\_i$ is a paranorm then so is $p$ and moreover, $p$ induces the same topology on $X$ as the family $p\_\{\backslash bull\}$ of paranorms. This is also true of the following paranorms on $X$:

• $q(x)\; :=\; \backslash inf\_\{\}\; \backslash left\backslash \{\; \backslash sum\_\{i=1\}^n\; p\_i(x)\; +\; \backslash frac\{1\}\{n\}\; ~:~\; n\; >\; 0\; \backslash text\{\; is\; an\; integer\; \}\backslash right\backslash \}.$
• $r(x)\; :=\; \backslash sum\_\{n=1\}^\{\backslash infty\}\; \backslash min\; \backslash left\backslash \{\; \backslash frac\{1\}\{2^n\},\; p\_n(x)\backslash right\backslash \}.$

---

Generalization The Fréchet combination can be generalized by use of a bounded remetrization function.

A is a continuous non-negative non-decreasing map $R\; :\; [0,\; \backslash infty)\; \backslash to\; [0,\; \backslash infty)$ that has a bounded range, is Subadditivity (meaning that $R(s\; +\; t)\; \backslash leq\; R(s)\; +\; R(t)$ for all $s,\; t\; \backslash geq\; 0$), and satisfies $R(s)\; =\; 0$ if and only if $s\; =\; 0.$

Examples of bounded remetrization functions include $\backslash arctan\; t,$ $\backslash tanh\; t,$ $t\; \backslash mapsto\; \backslash min\; \backslash \{t,\; 1\backslash \},$ and $t\; \backslash mapsto\; \backslash frac\{t\}\{1\; +\; t\}.$ If $d$ is a pseudometric (respectively, metric) on $X$ and $R$ is a bounded remetrization function then $R\; \backslash circ\; d$ is a bounded pseudometric (respectively, bounded metric) on $X$ that is uniformly equivalent to $d.$

Suppose that $p\_\backslash bull\; =\; \backslash left(p\_i\backslash right)\_\{i=1\}^\backslash infty$ is a family of non-negative F-seminorm on a vector space $X,$ $R$ is a bounded remetrization function, and $r\_\backslash bull\; =\; \backslash left(r\_i\backslash right)\_\{i=1\}^\backslash infty$ is a sequence of positive real numbers whose sum is finite. Then $$p(x)\; :=\; \backslash sum\_\{i=1\}^\backslash infty\; r\_i\; R\backslash left(p\_i(x)\backslash right)$$ defines a bounded F-seminorm that is uniformly equivalent to the $p\_\backslash bull.$ It has the property that for any net $x\_\backslash bull\; =\; \backslash left(x\_a\backslash right)\_\{a\; \backslash in\; A\}$ in $X,$ $p\backslash left(x\_\backslash bull\backslash right)\; \backslash to\; 0$ if and only if $p\_i\backslash left(x\_\backslash bull\backslash right)\; \backslash to\; 0$ for all $i.$ $p$ is an F-norm if and only if the $p\_\backslash bull$ separate points on $X.$

---

Characterizations

---

Of (pseudo)metrics induced by (semi)norms A pseudometric (resp. metric) $d$ is induced by a seminorm (resp. norm) on a vector space $X$ if and only if $d$ is translation invariant and absolutely homogeneous, which means that for all scalars $s$ and all $x,\; y\; \backslash in\; X,$ in which case the function defined by $p(x)\; :=\; d(x,\; 0)$ is a seminorm (resp. norm) and the pseudometric (resp. metric) induced by $p$ is equal to $d.$

---

Of pseudometrizable TVS If $(X,\; \backslash tau)$ is a topological vector space (TVS) (where note in particular that $\backslash tau$ is assumed to be a vector topology) then the following are equivalent:

1. $X$ is pseudometrizable (i.e. the vector topology $\backslash tau$ is induced by a pseudometric on $X$).
2. $X$ has a countable neighborhood base at the origin.
3. The topology on $X$ is induced by a translation-invariant pseudometric on $X.$
4. The topology on $X$ is induced by an F-seminorm.
5. The topology on $X$ is induced by a paranorm.

---

Of metrizable TVS If $(X,\; \backslash tau)$ is a TVS then the following are equivalent:

1. $X$ is metrizable.
2. $X$ is Hausdorff space and pseudometrizable.
3. $X$ is Hausdorff and has a countable neighborhood base at the origin.
4. The topology on $X$ is induced by a translation-invariant metric on $X.$
5. The topology on $X$ is induced by an F-norm.
6. The topology on $X$ is induced by a monotone F-norm.
7. The topology on $X$ is induced by a total paranorm.

---

Of locally convex pseudometrizable TVS If $(X,\; \backslash tau)$ is TVS then the following are equivalent:

1. $X$ is locally convex and pseudometrizable.
2. $X$ has a countable neighborhood base at the origin consisting of convex sets.
3. The topology of $X$ is induced by a countable family of (continuous) seminorms.
4. The topology of $X$ is induced by a countable increasing sequence of (continuous) seminorms $\backslash left(p\_i\backslash right)\_\{i=1\}^\{\backslash infty\}$ (increasing means that for all $i,$ $p\_i\; \backslash geq\; p\_\{i+1\}.$
5. The topology of $X$ is induced by an F-seminorm of the form: $$p(x)\; =\; \backslash sum\_\{n=1\}^\{\backslash infty\}\; 2^\{-n\}\; \backslash operatorname\{arctan\}\; p\_n(x)$$ where $\backslash left(p\_i\backslash right)\_\{i=1\}^\{\backslash infty\}$ are (continuous) seminorms on $X.$

---

Quotients Let $M$ be a vector subspace of a topological vector space $(X,\; \backslash tau).$

• If $X$ is a pseudometrizable TVS then so is $X\; /\; M.$
• If $X$ is a complete pseudometrizable TVS and $M$ is a closed vector subspace of $X$ then $X\; /\; M$ is complete.
• If $X$ is metrizable TVS and $M$ is a closed vector subspace of $X$ then $X\; /\; M$ is metrizable.
• If $p$ is an F-seminorm on $X,$ then the map $P\; :\; X\; /\; M\; \backslash to\; \backslash R$ defined by $$P(x\; +\; M)\; :=\; \backslash inf\_\{\}\; \backslash \{\; p(x\; +\; m)\; :\; m\; \backslash in\; M\; \backslash \}$$ is an F-seminorm on $X\; /\; M$ that induces the usual quotient topology on $X\; /\; M.$ If in addition $p$ is an F-norm on $X$ and if $M$ is a closed vector subspace of $X$ then $P$ is an F-norm on $X.$

---

Examples and sufficient conditions

• Every seminormed space $(X,\; p)$ is pseudometrizable with a canonical pseudometric given by $d(x,\; y)\; :=\; p(x\; -\; y)$ for all $x,\; y\; \backslash in\; X.$.
• If $(X,\; d)$ is pseudometric TVS with a translation invariant pseudometric $d,$ then $p(x)\; :=\; d(x,\; 0)$ defines a paranorm. However, if $d$ is a translation invariant pseudometric on the vector space $X$ (without the addition condition that $(X,\; d)$ is ), then $d$ need not be either an F-seminorm nor a paranorm.
• If a TVS has a bounded neighborhood of the origin then it is pseudometrizable; the converse is in general false.
• If a Hausdorff TVS has a bounded neighborhood of the origin then it is metrizable.
• Suppose $X$ is either a DF-space or an LM-space. If $X$ is a sequential space then it is either metrizable or else a Montel space DF-space.

If $X$ is Hausdorff locally convex TVS then $X$ with the strong topology, $\backslash left(X,\; b\backslash left(X,\; X^\{\backslash prime\}\backslash right)\backslash right),$ is metrizable if and only if there exists a countable set $\backslash mathcal\{B\}$ of bounded subsets of $X$ such that every bounded subset of $X$ is contained in some element of $\backslash mathcal\{B\}.$

The strong dual space $X\_b^\{\backslash prime\}$ of a metrizable locally convex space (such as a Fréchet spaceGabriyelyan, S.S. "On topological spaces and topological groups with certain local countable networks (2014)) $X$ is a DF-space. The strong dual of a DF-space is a Fréchet space. The strong dual of a Reflexive space Fréchet space is a bornological space. The strong bidual (that is, the strong dual space of the strong dual space) of a metrizable locally convex space is a Fréchet space. If $X$ is a metrizable locally convex space then its strong dual $X\_b^\{\backslash prime\}$ has one of the following properties, if and only if it has all of these properties: (1) bornological, (2) infrabarreled, (3) Barrelled space.

---

Normability A topological vector space is Seminormed space if and only if it has a Convex set bounded neighborhood of the origin. Moreover, a TVS is Normable space if and only if it is Hausdorff space and seminormable. Every metrizable TVS on a finite-Hamel basis vector space is a normable locally convex complete TVS, being TVS isomorphism to Euclidean space. Consequently, any metrizable TVS that is normable must be infinite dimensional.

If $M$ is a metrizable locally convex TVS that possess a Countable set fundamental system of bounded sets, then $M$ is normable.

If $X$ is a Hausdorff locally convex space then the following are equivalent:

1. $X$ is Normable space.
2. $X$ has a (von Neumann) bounded neighborhood of the origin.
3. the strong dual space $X^\{\backslash prime\}\_b$ of $X$ is normable.

and if this locally convex space $X$ is also metrizable, then the following may be appended to this list:

4. the strong dual space of $X$ is metrizable.
5. the strong dual space of $X$ is a Fréchet–Urysohn locally convex space.

In particular, if a metrizable locally convex space $X$ (such as a Fréchet space) is normable then its strong dual space $X^\{\backslash prime\}\_b$ is not a Fréchet–Urysohn space and consequently, this complete Hausdorff locally convex space $X^\{\backslash prime\}\_b$ is also neither metrizable nor normable.

Another consequence of this is that if $X$ is a Reflexive space locally convex TVS whose strong dual $X^\{\backslash prime\}\_b$ is metrizable then $X^\{\backslash prime\}\_b$ is necessarily a reflexive Fréchet space, $X$ is a DF-space, both $X$ and $X^\{\backslash prime\}\_b$ are necessarily complete Hausdorff ultrabornological distinguished , and moreover, $X^\{\backslash prime\}\_b$ is normable if and only if $X$ is normable if and only if $X$ is Fréchet–Urysohn if and only if $X$ is metrizable. In particular, such a space $X$ is either a Banach space or else it is not even a Fréchet–Urysohn space.

---

Metrically bounded sets and bounded sets Suppose that $(X,\; d)$ is a pseudometric space and $B\; \backslash subseteq\; X.$ The set $B$ is metrically bounded or $d$-bounded if there exists a real number $R\; >\; 0$ such that $d(x,\; y)\; \backslash leq\; R$ for all $x,\; y\; \backslash in\; B$; the smallest such $R$ is then called the diameter or $d$-diameter of $B.$ If $B$ is bounded in a pseudometrizable TVS $X$ then it is metrically bounded; the converse is in general false but it is true for locally convex metrizable TVSs.

---

Properties of pseudometrizable TVS

• Every metrizable locally convex TVS is a quasibarrelled space, bornological space, and a Mackey space.
• Every complete metrizable TVS is a barrelled space and a Baire space (and hence non-meager). However, there exist metrizable Baire spaces that are not complete.
• If $X$ is a metrizable locally convex space, then the strong dual of $X$ is bornological if and only if it is barreled space, if and only if it is infrabarreled.
• If $X$ is a complete pseudometrizable TVS and $M$ is a closed vector subspace of $X,$ then $X\; /\; M$ is complete.
• The strong dual of a locally convex metrizable TVS is a webbed space.
• If $(X,\; \backslash tau)$ and $(X,\; \backslash nu)$ are complete metrizable TVSs (i.e. ) and if $\backslash nu$ is coarser than $\backslash tau$ then $\backslash tau\; =\; \backslash nu$; this is no longer guaranteed to be true if any one of these metrizable TVSs is not complete. Said differently, if $(X,\; \backslash tau)$ and $(X,\; \backslash nu)$ are both but with different topologies, then neither one of $\backslash tau$ and $\backslash nu$ contains the other as a subset. One particular consequence of this is, for example, that if $(X,\; p)$ is a Banach space and $(X,\; q)$ is some other normed space whose norm-induced topology is finer than (or alternatively, is coarser than) that of $(X,\; p)$ (i.e. if $p\; \backslash leq\; C\; q$ or if $q\; \backslash leq\; C\; p$ for some constant $C\; >\; 0$), then the only way that $(X,\; q)$ can be a Banach space (i.e. also be complete) is if these two norms $p$ and $q$ are Equivalent norms; if they are not equivalent, then $(X,\; q)$ can not be a Banach space. As another consequence, if $(X,\; p)$ is a Banach space and $(X,\; \backslash nu)$ is a Fréchet space, then the map $p\; :\; (X,\; \backslash nu)\; \backslash to\; \backslash R$ is continuous if and only if the Fréchet space $(X,\; \backslash nu)$ the TVS $(X,\; p)$ (here, the Banach space $(X,\; p)$ is being considered as a TVS, which means that its norm is "forgetten" but its topology is remembered).
• A metrizable locally convex space is Normable space if and only if its strong dual space is a Fréchet–Urysohn locally convex space.
• Any product of complete metrizable TVSs is a Baire space.
• A product of metrizable TVSs is metrizable if and only if it all but at most countably many of these TVSs have dimension $0.$
• A product of pseudometrizable TVSs is pseudometrizable if and only if it all but at most countably many of these TVSs have the trivial topology.
• Every complete metrizable TVS is a barrelled space and a Baire space (and thus non-meager).
• The dimension of a complete metrizable TVS is either finite or uncountable.

---

Completeness Every topological vector space (and more generally, a topological group) has a canonical uniform structure, induced by its topology, which allows the notions of completeness and uniform continuity to be applied to it. If $X$ is a metrizable TVS and $d$ is a metric that defines $X$'s topology, then its possible that $X$ is complete as a TVS (i.e. relative to its uniformity) but the metric $d$ is a complete metric (such metrics exist even for $X\; =\; \backslash R$). Thus, if $X$ is a TVS whose topology is induced by a pseudometric $d,$ then the notion of completeness of $X$ (as a TVS) and the notion of completeness of the pseudometric space $(X,\; d)$ are not always equivalent. The next theorem gives a condition for when they are equivalent:

If $M$ is a closed vector subspace of a complete pseudometrizable TVS $X,$ then the quotient space $X\; /\; M$ is complete. If $M$ is a vector subspace of a metrizable TVS $X$ and if the quotient space $X\; /\; M$ is complete then so is $X.$ If $X$ is not complete then $M\; :=\; X,$ but not complete, vector subspace of $X.$

A Baire space Separable space topological group is metrizable if and only if it is cosmic.Gabriyelyan, S.S. "On topological spaces and topological groups with certain local countable networks (2014)

---

Subsets and subsequences

• Let $M$ be a Separable space locally convex metrizable topological vector space and let $C$ be its completion. If $S$ is a bounded subset of $C$ then there exists a bounded subset $R$ of $X$ such that $S\; \backslash subseteq\; \backslash operatorname\{cl\}\_C\; R.$
• Every totally bounded subset of a locally convex metrizable TVS $X$ is contained in the closed convex balanced hull of some sequence in $X$ that converges to $0.$
• In a pseudometrizable TVS, every bornivore is a neighborhood of the origin.
• If $d$ is a translation invariant metric on a vector space $X,$ then $d(n\; x,\; 0)\; \backslash leq\; n\; d(x,\; 0)$ for all $x\; \backslash in\; X$ and every positive integer $n.$
• If $\backslash left(x\_i\backslash right)\_\{i=1\}^\{\backslash infty\}$ is a null sequence (that is, it converges to the origin) in a metrizable TVS then there exists a sequence $\backslash left(r\_i\backslash right)\_\{i=1\}^\{\backslash infty\}$ of positive real numbers diverging to $\backslash infty$ such that $\backslash left(r\_i\; x\_i\backslash right)\_\{i=1\}^\{\backslash infty\}\; \backslash to\; 0.$
• A subset of a complete metric space is closed if and only if it is complete. If a space $X$ is not complete, then $X$ is a closed subset of $X$ that is not complete.
• If $X$ is a metrizable locally convex TVS then for every bounded subset $B$ of $X,$ there exists a bounded disk $D$ in $X$ such that $B\; \backslash subseteq\; X\_D,$ and both $X$ and the auxiliary normed space $X\_D$ induce the same subspace topology on $B.$

Generalized series

As described in this article's section on generalized series, for any $I$-Indexing set family $\backslash left(r\_i\backslash right)\_\{i\; \backslash in\; I\}$ of vectors from a TVS $X,$ it is possible to define their sum $\backslash textstyle\backslash sum\backslash limits\_\{i\; \backslash in\; I\}\; r\_i$ as the limit of the net of finite partial sums $F\; \backslash in\; \backslash operatorname\{FiniteSubsets\}(I)\; \backslash mapsto\; \backslash textstyle\backslash sum\backslash limits\_\{i\; \backslash in\; F\}\; r\_i$ where the domain $\backslash operatorname\{FiniteSubsets\}(I)$ is Directed set by $\backslash ,\backslash subseteq.\backslash ,$ If $I\; =\; \backslash N$ and $X\; =\; \backslash Reals,$ for instance, then the generalized series $\backslash textstyle\backslash sum\backslash limits\_\{i\; \backslash in\; \backslash N\}\; r\_i$ converges if and only if $\backslash textstyle\backslash sum\backslash limits\_\{i=1\}^\backslash infty\; r\_i$ converges unconditionally in the usual sense (which for real numbers, is equivalent to absolute convergence). If a generalized series $\backslash textstyle\backslash sum\backslash limits\_\{i\; \backslash in\; I\}\; r\_i$ converges in a metrizable TVS, then the set $\backslash left\backslash \{i\; \backslash in\; I\; :\; r\_i\; \backslash neq\; 0\backslash right\backslash \}$ is necessarily Countable set (that is, either finite or countably infinite); in other words, all but at most countably many $r\_i$ will be zero and so this generalized series $\backslash textstyle\backslash sum\backslash limits\_\{i\; \backslash in\; I\}\; r\_i\; ~=~\; \backslash textstyle\backslash sum\backslash limits\_\{\backslash stackrel\{i\; \backslash in\; I\}\{r\_i\; \backslash neq\; 0\}\}\; r\_i$ is actually a sum of at most countably many non-zero terms.

---

Linear maps If $X$ is a pseudometrizable TVS and $A$ maps bounded subsets of $X$ to bounded subsets of $Y,$ then $A$ is continuous. Discontinuous linear functionals exist on any infinite-dimensional pseudometrizable TVS. Thus, a pseudometrizable TVS is finite-dimensional if and only if its continuous dual space is equal to its algebraic dual space.

If $F\; :\; X\; \backslash to\; Y$ is a linear map between TVSs and $X$ is metrizable then the following are equivalent:

1. $F$ is continuous;
2. $F$ is a (locally) bounded map (that is, $F$ maps (von Neumann) bounded subsets of $X$ to bounded subsets of $Y$);
3. $F$ is sequentially continuous;
4. the image under $F$ of every null sequence in $X$ is a bounded set where by definition, a is a sequence that converges to the origin.
5. $F$ maps null sequences to null sequences;

Open and almost open maps

Theorem: If $X$ is a complete pseudometrizable TVS, $Y$ is a Hausdorff TVS, and $T\; :\; X\; \backslash to\; Y$ is a closed and almost open linear surjection, then $T$ is an open map.

Theorem: If $T\; :\; X\; \backslash to\; Y$ is a surjective linear operator from a locally convex space $X$ onto a barrelled space $Y$ (e.g. every complete pseudometrizable space is barrelled) then $T$ is almost open.

Theorem: If $T\; :\; X\; \backslash to\; Y$ is a surjective linear operator from a TVS $X$ onto a Baire space $Y$ then $T$ is almost open.

Theorem: Suppose $T\; :\; X\; \backslash to\; Y$ is a continuous linear operator from a complete pseudometrizable TVS $X$ into a Hausdorff TVS $Y.$ If the image of $T$ is non-Meager set in $Y$ then $T\; :\; X\; \backslash to\; Y$ is a surjective open map and $Y$ is a complete metrizable space.

---

Hahn-Banach extension property A vector subspace $M$ of a TVS $X$ has the extension property if any continuous linear functional on $M$ can be extended to a continuous linear functional on $X.$ Say that a TVS $X$ has the Hahn-Banach extension property ( HBEP) if every vector subspace of $X$ has the extension property.

The Hahn-Banach theorem guarantees that every Hausdorff locally convex space has the HBEP. For complete metrizable TVSs there is a converse:

If a vector space $X$ has uncountable dimension and if we endow it with the finest vector topology then this is a TVS with the HBEP that is neither locally convex or metrizable.

---

See also

---

Notes Proofs

{=}~ {\textstyle\lim\limits_{A \in \operatorname{FiniteSubsets}(I)}} \ \textstyle\sum\limits_{i \in A} r_i = \lim \left\{\textstyle\sum\limits_{i\in A} r_i \,: A \subseteq I, A \text{ finite }\right\} converges to some point in a metrizable TVS $X,$ where recall that this net's domain is the directed set $(\backslash operatorname\{FiniteSubsets\}(I),\; \backslash subseteq).$ Like every convergent net, this convergent net of partial sums $A\; \backslash mapsto\; \backslash textstyle\backslash sum\backslash limits\_\{i\; \backslash in\; A\}\; r\_i$ is a , which for this particular net means (by definition) that for every neighborhood $W$ of the origin in $X,$ there exists a finite subset $A\_0$ of $I$ such that $\backslash textstyle\backslash sum\backslash limits\_\{i\; \backslash in\; B\}\; r\_i\; -\; \backslash textstyle\backslash sum\backslash limits\_\{i\; \backslash in\; C\}\; r\_i\; \backslash in\; W$ for all finite supersets $B,\; C\; \backslash supseteq\; A\_0;$ this implies that $r\_i\; \backslash in\; W$ for every $i\; \backslash in\; I\; \backslash setminus\; A\_0$ (by taking $B\; :=\; A\_0\; \backslash cup\; \backslash \{i\backslash \}$ and $C\; :=\; A\_0$). Since $X$ is metrizable, it has a countable neighborhood basis $U\_1,\; U\_2,\; \backslash ldots$ at the origin, whose intersection is necessarily $U\_1\; \backslash cap\; U\_2\; \backslash cap\; \backslash cdots\; =\; \backslash \{0\backslash \}$ (since $X$ is a Hausdorff TVS). For every positive integer $n\; \backslash in\; \backslash N,$ pick a finite subset $A\_n\; \backslash subseteq\; I$ such that $r\_i\; \backslash in\; U\_n$ for every $i\; \backslash in\; I\; \backslash setminus\; A\_n.$ If $i$ belongs to $(I\; \backslash setminus\; A\_1)\; \backslash cap\; (I\; \backslash setminus\; A\_2)\; \backslash cap\; \backslash cdots\; =\; I\; \backslash setminus\; \backslash left(A\_1\; \backslash cup\; A\_2\; \backslash cup\; \backslash cdots\backslash right)$ then $r\_i$ belongs to $U\_1\; \backslash cap\; U\_2\; \backslash cap\; \backslash cdots\; =\; \backslash \{0\backslash \}.$ Thus $r\_i\; =\; 0$ for every index $i\; \backslash in\; I$ that does not belong to the countable set $A\_1\; \backslash cup\; A\_2\; \backslash cup\; \backslash cdots.$ $\backslash blacksquare$

---

Bibliography

• Taqdir Husain (1978). 9783540090960, Springer-Verlag. ISBN 9783540090960

Post Comment