Banach space

 ( Functional Analysis ) 

The weak topology of a Banach space $X$ is metrizable if and only if $X$ is finite-dimensional.see Proposition 2.5.14, p. 215 in . If the dual $X\text{'}$ is separable, the weak topology of the unit ball of $X$ is metrizable. This applies in particular to separable reflexive Banach spaces. Although the weak topology of the unit ball is not metrizable in general, one can characterize weak compactness using sequences.

A Banach space $X$ is reflexive if and only if each bounded sequence in $X$ has a weakly convergent subsequence.see Corollary 2.8.9, p. 251 in .

A weakly compact subset $A$ in $\backslash ell^1$ is norm-compact. Indeed, every sequence in $A$ has weakly convergent subsequences by Eberlein–Šmulian, that are norm convergent by the Schur property of $\backslash ell^1.$

Banach spaces with a Schauder basis are necessarily Separable space, because the countable set of finite linear combinations with rational coefficients (say) is dense.

It follows from the Banach–Steinhaus theorem that the linear mappings $\backslash \{P\_n\backslash \}$ are uniformly bounded by some constant $C.$ Let $\backslash \{e\_n^*\backslash \}$ denote the coordinate functionals which assign to every $x$ in $X$ the coordinate $x\_n$ of $x$ in the above expansion. They are called biorthogonal functionals. When the basis vectors have norm $1,$ the coordinate functionals $\backslash \{e\_n^*\backslash \}$ have norm $\{\}\backslash leq\; 2\; C$ in the dual of $X.$

Most classical separable spaces have explicit bases. The Haar wavelet $\backslash \{h\_n\backslash \}$ is a basis for $L^p(0,)$ when The trigonometric system is a basis in $L^p(\backslash mathbf\{T\})$ when The Schauder system is a basis in the space $C(0,).$see p. 3. The question of whether the disk algebra $A(\backslash mathbf\{D\})$ has a basisthe question appears p. 238, §3 in Banach's book, . remained open for more than forty years, until Bočkarev showed in 1974 that $A(\backslash mathbf\{D\})$ admits a basis constructed from the Franklin system.see S. V. Bočkarev, "Existence of a basis in the space of functions analytic in the disc, and some properties of Franklin's system". (Russian) Mat. Sb. (N.S.) 95(137) (1974), 3–18, 159.

Since every vector $x$ in a Banach space $X$ with a basis is the limit of $P\_n(x),$ with $P\_n$ of finite rank and uniformly bounded, the space $X$ satisfies the bounded approximation property. The first example by Per Enflo of a space failing the approximation property was at the same time the first example of a separable Banach space without a Schauder basis.see

Robert C. James characterized reflexivity in Banach spaces with a basis: the space $X$ with a Schauder basis is reflexive if and only if the basis is both shrinking and boundedly complete.see R.C. James, "Bases and reflexivity of Banach spaces". Ann. of Math. (2) 52, (1950). 518–527. See also p. 9. In this case, the biorthogonal functionals form a basis of the dual of $X.$

If $T\_1\; :\; X\; \backslash times\; Y\; \backslash to\; Z\_1$ is any bilinear mapping into a $\backslash mathbb\{K\}$-vector space $Z\_1,$ then there exists a unique linear mapping $f\; :\; Z\; \backslash to\; Z\_1$ such that $T\_1\; =\; f\; \backslash circ\; T.$

The image under $T$ of a couple $(x,\; y)$ in $X\; \backslash times\; Y$ is denoted by $x\; \backslash otimes\; y,$ and called a simple tensor. Every element $z$ in $X\; \backslash otimes\; Y$ is a finite sum of such simple tensors.

There are various norms that can be placed on the tensor product of the underlying vector spaces, amongst others the projective cross norm and injective cross norm introduced by A. Grothendieck in 1955.see A. Grothendieck, "Produits tensoriels topologiques et espaces nucléaires". Mem. Amer. Math. Soc. 1955 (1955), no. 16, 140 pp., and A. Grothendieck, "Résumé de la théorie métrique des produits tensoriels topologiques". Bol. Soc. Mat. São Paulo 8 1953 1–79.

In general, the tensor product of complete spaces is not complete again. When working with Banach spaces, it is customary to say that the projective tensor productsee chap. 2, p. 15 in . of two Banach spaces $X$ and $Y$ is the $X\; \backslash widehat\{\backslash otimes\}\_\backslash pi\; Y$ of the algebraic tensor product $X\; \backslash otimes\; Y$ equipped with the projective tensor norm, and similarly for the injective tensor productsee chap. 3, p. 45 in . $X\; \backslash widehat\{\backslash otimes\}\_\backslash varepsilon\; Y.$ Grothendieck proved in particular thatsee Example. 2.19, p. 29, and pp. 49–50 in .

where $K$ is a compact Hausdorff space, $C(K,\; Y)$ the Banach space of continuous functions from $K$ to $Y$ and $L^1(0,,\; Y)$ the space of Bochner-measurable and integrable functions from $0,$ to $Y,$ and where the isomorphisms are isometric. The two isomorphisms above are the respective extensions of the map sending the tensor $f\; \backslash otimes\; y$ to the vector-valued function $s\; \backslash in\; K\; \backslash to\; f(s)\; y\; \backslash in\; Y.$

For every Banach space $Y,$ there is a natural norm $1$ linear map $$Y\; \backslash widehat\backslash otimes\_\backslash pi\; X\; \backslash to\; Y\; \backslash widehat\backslash otimes\_\backslash varepsilon\; X$$ obtained by extending the identity map of the algebraic tensor product. Grothendieck related the approximation problem to the question of whether this map is one-to-one when $Y$ is the dual of $X.$ Precisely, for every Banach space $X,$ the map $$X\text{'}\; \backslash widehat\; \backslash otimes\_\backslash pi\; X\; \backslash \; \backslash longrightarrow\; X\text{'}\; \backslash widehat\; \backslash otimes\_\backslash varepsilon\; X$$ is one-to-one if and only if $X$ has the approximation property.see Proposition 4.6, p. 74 in .

Grothendieck conjectured that $X\; \backslash widehat\{\backslash otimes\}\_\backslash pi\; Y$ and $X\; \backslash widehat\{\backslash otimes\}\_\backslash varepsilon\; Y$ must be different whenever $X$ and $Y$ are infinite-dimensional Banach spaces. This was disproved by Gilles Pisier in 1983.see Pisier, Gilles (1983), "Counterexamples to a conjecture of Grothendieck", Acta Math. 151:181–208. Pisier constructed an infinite-dimensional Banach space $X$ such that $X\; \backslash widehat\{\backslash otimes\}\_\backslash pi\; X$ and $X\; \backslash widehat\{\backslash otimes\}\_\backslash varepsilon\; X$ are equal. Furthermore, just as Per Enflo example, this space $X$ is a "hand-made" space that fails to have the approximation property. On the other hand, Szankowski proved that the classical space $B(\backslash ell^2)$ does not have the approximation property.see Szankowski, Andrzej (1981), "$B(H)$ does not have the approximation property", Acta Math. 147: 89–108. Ryan claims that this result is due to Per Enflo, p. 74 in .

It follows, for example, that the Lp space $L^p(0,)$ is a Hilbert space only when $p\; =\; 2.$ If this identity is satisfied, the associated inner product is given by the polarization identity. In the case of real scalars, this gives: $$\backslash langle\; x,\; y\backslash rangle\; =\; \backslash tfrac\{1\}\{4\}(\backslash |x+y\backslash |^2\; -\; \backslash |x-y\backslash |^2).$$

For complex scalars, defining the inner product so as to be $\backslash Complex$-linear in $x,$ Antilinear map in $y,$ the polarization identity gives: $$\backslash langle\; x,y\backslash rangle\; =\; \backslash tfrac\{1\}\{4\}\backslash left(\backslash |x+y\backslash |^2\; -\; \backslash |x-y\backslash |^2\; +\; i(\backslash |x+iy\backslash |^2\; -\; \backslash |x-iy\backslash |^2)\backslash right).$$

To see that the parallelogram law is sufficient, one observes in the real case that $\backslash langle\; x,\; y\; \backslash rangle$ is symmetric, and in the complex case, that it satisfies the Hermitian symmetry property and $\backslash langle\; i\; x,\; y\; \backslash rangle\; =\; i\; \backslash langle\; x,\; y\; \backslash rangle.$ The parallelogram law implies that $\backslash langle\; x,\; y\; \backslash rangle$ is additive in $x.$ It follows that it is linear over the rationals, thus linear by continuity.

Several characterizations of spaces isomorphic (rather than isometric) to Hilbert spaces are available. The parallelogram law can be extended to more than two vectors, and weakened by the introduction of a two-sided inequality with a constant $c\; \backslash geq\; 1$: Kwapień proved that if $$c^\{-2\}\; \backslash sum\_\{k=1\}^n\; \backslash |x\_k\backslash |^2\; \backslash leq\; \backslash operatorname\{Ave\}\_\{\backslash pm\}\; \backslash left\backslash |\backslash sum\_\{k=1\}^n\; \backslash pm\; x\_k\backslash right\backslash |^2\; \backslash leq\; c^2\; \backslash sum\_\{k=1\}^n\; \backslash |x\_k\backslash |^2$$ for every integer $n$ and all families of vectors $\backslash \{x\_1,\; \backslash ldots,\; x\_n\backslash \}\; \backslash subseteq\; X,$ then the Banach space $X$ is isomorphic to a Hilbert space.see Kwapień, S. (1970), "A linear topological characterization of inner-product spaces", Studia Math. 38:277–278. Here, $\backslash operatorname\{Ave\}\_\{\backslash pm\}$ denotes the average over the $2^n$ possible choices of signs $\backslash pm\; 1.$ In the same article, Kwapień proved that the validity of a Banach-valued Parseval's theorem for the Fourier transform characterizes Banach spaces isomorphic to Hilbert spaces.

Lindenstrauss and Tzafriri proved that a Banach space in which every closed linear subspace is complemented (that is, is the range of a bounded linear projection) is isomorphic to a Hilbert space. The proof rests upon Dvoretzky's theorem about Euclidean sections of high-dimensional centrally symmetric convex bodies. In other words, Dvoretzky's theorem states that for every integer $n,$ any finite-dimensional normed space, with dimension sufficiently large compared to $n,$ contains subspaces nearly isometric to the $n$-dimensional Euclidean space.

The next result gives the solution of the so-called . An infinite-dimensional Banach space $X$ is said to be homogeneous if it is isomorphic to all its infinite-dimensional closed subspaces. A Banach space isomorphic to $\backslash ell^2$ is homogeneous, and Banach asked for the converse.see p. 245 in . The homogeneity property is called "propriété (15)" there. Banach writes: "on ne connaît aucun exemple d'espace à une infinité de dimensions qui, sans être isomorphe avec $(L^2).$ possède la propriété (15)".

An infinite-dimensional Banach space is hereditarily indecomposable when no subspace of it can be isomorphic to the direct sum of two infinite-dimensional Banach spaces. The Timothy Gowers dichotomy theorem asserts that every infinite-dimensional Banach space $X$ contains, either a subspace $Y$ with unconditional basis, or a hereditarily indecomposable subspace $Z,$ and in particular, $Z$ is not isomorphic to its closed hyperplanes.see If $X$ is homogeneous, it must therefore have an unconditional basis. It follows then from the partial solution obtained by Komorowski and Tomczak–Jaegermann, for spaces with an unconditional basis,see and also that $X$ is isomorphic to $\backslash ell^2.$

Anderson–Kadec theorem (1965–66) proves that any two infinite-dimensional separable space Banach spaces are homeomorphic as topological spaces. Kadec's theorem was extended by Torunczyk, who proved that any two Banach spaces are homeomorphic if and only if they have the same density character, the minimum cardinality of a dense subset.

The situation is different for Countable set compact Hausdorff spaces. Every countably infinite compact $K$ is homeomorphic to some closed interval of $$\backslash langle\; 1,\; \backslash alpha\; \backslash rangle\; =\; \backslash \{\; \backslash gamma\; \backslash mid\; 1\; \backslash leq\; \backslash gamma\; \backslash leq\; \backslash alpha\backslash \}$$ equipped with the order topology, where $\backslash alpha$ is a countably infinite ordinal.One can take , where $\backslash beta\; +\; 1$ is the Cantor–Bendixson rank of $K,$ and $n\; >\; 0$ is the finite number of points in the $\backslash beta$-th derived set $K(\backslash beta)$ of $K.$ See Mazurkiewicz, Stefan; Sierpiński, Wacław (1920), "Contribution à la topologie des ensembles dénombrables", Fundamenta Mathematicae 1: 17–27. The Banach space $C(K)$ is then isometric to . When $\backslash alpha,\; \backslash beta$ are two countably infinite ordinals, and assuming $\backslash alpha\; \backslash leq\; \backslash beta,$ the spaces and are isomorphic if and only if .Bessaga, Czesław; Pełczyński, Aleksander (1960), "Spaces of continuous functions. IV. On isomorphical classification of spaces of continuous functions", Studia Math. 19:53–62. For example, the Banach spaces $$C(\backslash langle\; 1,\; \backslash omega\backslash rangle),\; \backslash \; C(\backslash langle\; 1,\; \backslash omega^\{\backslash omega\}\; \backslash rangle),\; \backslash \; C(\backslash langle\; 1,\; \backslash omega^\{\backslash omega^2\}\backslash rangle),\; \backslash \; C(\backslash langle\; 1,\; \backslash omega^\{\backslash omega^3\}\; \backslash rangle),\; \backslash cdots,\; C(\backslash langle\; 1,\; \backslash omega^\{\backslash omega^\backslash omega\}\; \backslash rangle),\; \backslash cdots$$ are mutually non-isomorphic.

• .*
• .
• .
• .
• .
• .
• .
• .