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Banach Spaces

Field: Functional Analysis

Complete normed vector spaces.

Sequence of Expressions

Definition

Definition

A Banach space is a completenormed space (X,∥⋅∥).(X,\|{\cdot }\|). A normed space is a pair (X,∥⋅∥)(X,\|{\cdot }\|) consisting of a vector space XX over a scalar field K\mathbb {K} (where K\mathbb {K} is commonly R\mathbb {R} or C\mathbb {C} ) together with a distinguishednorm ∥⋅∥:X→R.\|{\cdot }\|:X\to \mathbb {R} . Like all norms, this norm induces a translation invariantdistance function, called the canonical or (norm) induced metric, defined for all vectors x,y∈Xx,y\in X by d(x,y):=∥y−x∥=∥x−y∥.d(x,y):=\|y-x\|=\|x-y\|. This makes XX into a metric space (X,d).(X,d). A sequence x1,x2,…x_{1},x_{2},\ldots is called Cauchy in (X,d)(X,d) or dd -Cauchy or ∥⋅∥\|{\cdot }\| -Cauchy if for every real r>0,r>0, there exists some index NN such that d(xn,xm)=∥xn−xm∥<rd(x_{n},x_{m})=\|x_{n}-x_{m}\|<r whenever mm and nn are greater than N.N. The normed space (X,∥⋅∥)(X,\|{\cdot }\|) is called a Banach space and the canonical metric dd is called a complete metric if (X,d)(X,d) is a complete metric space, which by definition means for every Cauchy sequence x1,x2,…x_{1},x_{2},\ldots in (X,d),(X,d), there exists some x∈Xx\in X such that lim⁡n→∞xn=x   in (X,d),\lim _{n\to \infty }x_{n}=x\;{\text{ in }}(X,d), where because ∥xn−x∥=d(xn,x),\|x_{n}-x\|=d(x_{n},x), this sequence's convergence to xx can equivalently be expressed as lim⁡n→∞∥xn−x∥=0   in R.\lim _{n\to \infty }\|x_{n}-x\|=0\;{\text{ in }}\mathbb {R} . The norm ∥⋅∥\|{\cdot }\| of a normed space (X,∥⋅∥)(X,\|{\cdot }\|) is called a complete norm if (X,∥⋅∥)(X,\|{\cdot }\|) is a Banach space. For any normed space (X,∥⋅∥),(X,\|{\cdot }\|), there exists an L-semi-inner product ⟨⋅,⋅⟩\langle \cdot ,\cdot \rangle on XX such that ∥x∥=⟨x,x⟩{\textstyle \|x\|={\sqrt {\langle x,x\rangle }}} for all x∈X.x\in X. In general, there may be infinitely many L-semi-inner products that satisfy this condition and the proof of the existence of L-semi-inner products relies on the non-constructive Hahn–Banach theorem. L-semi-inner products are a generalization of inner products, which are what fundamentally distinguish Hilbert spaces from all other Banach spaces. This shows that all normed spaces (and hence all Banach spaces) can be considered as being generalizations of (pre-)Hilbert spaces. The vector space structure allows one to relate the behavior of Cauchy sequences to that of converging series of vectors. A normed space XX is a Banach space if and only if each absolutely convergent series in XX converges to a value that lies within X,X, symbolically ∑n=1∞∥vn∥<∞  ⟹  ∑n=1∞vn converges in X.\sum _{n=1}^{\infty }\|v_{n}\|<\infty \implies \sum _{n=1}^{\infty }v_{n}{\text{ converges in }}X. The canonical metric dd of a normed space (X,∥⋅∥)(X,\|{\cdot }\|) induces the usual metric topology τd\tau _{d} on X,X, which is referred to as the canonical or norm induced topology. Every normed space is automatically assumed to carry this Hausdorff topology, unless indicated otherwise. With this topology, every Banach space is a Baire space, although there exist normed spaces that are Baire but not Banach. The norm ∥⋅∥:X→R\|{\cdot }\|:X\to \mathbb {R} is always a continuous function with respect to the topology that it induces. The open and closed balls of radius r>0r>0 centered at a point x∈Xx\in X are, respectively, the sets Br(x):={z∈X∣∥z−x∥<r} and Cr(x):={z∈X∣∥z−x∥≤r}.B_{r}(x):=\{z\in X\mid \|z-x\|<r\}\qquad {\text{ and }}\qquad C_{r}(x):=\{z\in X\mid \|z-x\|\leq r\}. Any such ball is a convex and bounded subset of X,X, but a compact ball/neighborhood exists if and only if XX is finite-dimensional. In particular, no infinite–dimensional normed space can be locally compact or have the Heine–Borel property. If x0x_{0} is a vector and s≠0s\neq 0 is a scalar, then x0+s Br(x)=B∣s∣r(x0+sx) and x0+s Cr(x)=C∣s∣r(x0+sx).x_{0}+s\,B_{r}(x)=B_{|s|r}(x_{0}+sx)\qquad {\text{ and }}\qquad x_{0}+s\,C_{r}(x)=C_{|s|r}(x_{0}+sx). Using s=1s=1 shows that the norm-induced topology is translation invariant, which means that for any x∈Xx\in X and S⊆X,S\subseteq X, the subset SS is open (respectively, closed) in XX if and only if its translation x+S:={x+s∣s∈S}x+S:=\{x+s\mid s\in S\} is open (respectively, closed). Consequently, the norm induced topology is completely determined by any neighbourhood basis at the origin. Some common neighborhood bases at the origin include {Br(0)∣r>0},{Cr(0)∣r>0},{Brn(0)∣n∈N}, and {Crn(0)∣n∈N},\{B_{r}(0)\mid r>0\},\qquad \{C_{r}(0)\mid r>0\},\qquad \{B_{r_{n}}(0)\mid n\in \mathbb {N} \},\qquad {\text{ and }}\qquad \{C_{r_{n}}(0)\mid n\in \mathbb {N} \}, where r1,r2,…r_{1},r_{2},\ldots can be any sequence of positive real numbers that converges to 00 in R\mathbb {R} (common choices are rn:=1nr_{n}:={\tfrac {1}{n}} or rn:=1/2nr_{n}:=1/2^{n} ). So, for example, any open subset UU of XX can be written as a union U=⋃x∈IBrx(x)=⋃x∈Ix+Brx(0)=⋃x∈Ix+rx B1(0)U=\bigcup _{x\in I}B_{r_{x}}(x)=\bigcup _{x\in I}x+B_{r_{x}}(0)=\bigcup _{x\in I}x+r_{x}\,B_{1}(0) indexed by some subset I⊆U,I\subseteq U, where each rxr_{x} may be chosen from the aforementioned sequence r1,r2,….r_{1},r_{2},\ldots . (The open balls can also be replaced with closed balls, although the indexing set II and radii rxr_{x} may then also need to be replaced). Additionally, II can always be chosen to be countable if XX is a separable space, which by definition means that XX contains some countable dense subset. All finite–dimensional normed spaces are separable Banach spaces and any two Banach spaces of the same finite dimension are linearly homeomorphic. Every separable infinite–dimensional Hilbert space is linearly isometrically isomorphic to the separable Hilbert sequence space ℓ2(N)\ell ^{2}(\mathbb {N} ) with its usual norm ∥⋅∥2.\|{\cdot }\|_{2}. The Anderson–Kadec theorem states that every infinite–dimensional separable Fréchet space is homeomorphic to the product space ∏i∈NR{\textstyle \prod _{i\in \mathbb {N} }\mathbb {R} } of countably many copies of R\mathbb {R} (this homeomorphism need not be a linear map). Thus all infinite–dimensional separable Fréchet spaces are homeomorphic to each other (or said differently, their topology is unique up to a homeomorphism). Since every Banach space is a Fréchet space, this is also true of all infinite–dimensional separable Banach spaces, including ℓ2(N).\ell ^{2}(\mathbb {N} ). In fact, ℓ2(N)\ell ^{2}(\mathbb {N} ) is even homeomorphic to its own unit sphere {x∈ℓ2(N)∣∥x∥2=1},\{x\in \ell ^{2}(\mathbb {N} )\mid \|x\|_{2}=1\}, which stands in sharp contrast to finite–dimensional spaces (the Euclidean plane R2\mathbb {R} ^{2} is not homeomorphic to the unit circle, for instance). This pattern in homeomorphism classes extends to generalizations of metrizable (locally Euclidean) topological manifolds known as metric Banach manifolds, which are metric spaces that are around every point, locally homeomorphic to some open subset of a given Banach space (metric Hilbert manifolds and metric Fréchet manifolds are defined similarly). For example, every open subset UU of a Banach space XX is canonically a metric Banach manifold modeled on XX since the inclusion map U→XU\to X is an openlocal homeomorphism. Using Hilbert space microbundles, David Henderson showed in 1969 that every metric manifold modeled on a separable infinite–dimensional Banach (or Fréchet) space can be topologically embedded as an open subset of ℓ2(N)\ell ^{2}(\mathbb {N} ) and, consequently, also admits a unique smooth structure making it into a C∞C^{\infty } Hilbert manifold. There is a compact subset SS of ℓ2(N)\ell ^{2}(\mathbb {N} ) whose convex hull co⁡(S)\operatorname {co} (S) is not closed and thus also not compact. However, like in all Banach spaces, the closed convex hull co⁡‾S{\overline {\operatorname {co} }}S of this (and every other) compact subset will be compact. In a normed space that is not complete then it is in general not guaranteed that co⁡‾S{\overline {\operatorname {co} }}S will be compact whenever SS is; an example can even be found in a (non-complete) pre-Hilbert vector subspace of ℓ2(N).\ell ^{2}(\mathbb {N} ). This norm-induced topology also makes (X,τd)(X,\tau _{d}) into what is known as a topological vector space (TVS), which by definition is a vector space endowed with a topology making the operations of addition and scalar multiplication continuous. It is emphasized that the TVS (X,τd)(X,\tau _{d}) is only a vector space together with a certain type of topology; that is to say, when considered as a TVS, it is not associated with any particular norm or metric (both of which are "forgotten"). This Hausdorff TVS (X,τd)(X,\tau _{d}) is even locally convex because the set of all open balls centered at the origin forms a neighbourhood basis at the origin consisting of convex balanced open sets. This TVS is also normable, which by definition refers to any TVS whose topology is induced by some (possibly unknown) norm. Normable TVSs are characterized by being Hausdorff and having a boundedconvex neighborhood of the origin. All Banach spaces are barrelled spaces, which means that every barrel is neighborhood of the origin (all closed balls centered at the origin are barrels, for example) and guarantees that the Banach–Steinhaus theorem holds. The open mapping theorem implies that when τ1\tau _{1} and τ2\tau _{2} are topologies on XX that make both (X,τ1)(X,\tau _{1}) and (X,τ2)(X,\tau _{2}) into complete metrizable TVSes (for example, Banach or Fréchet spaces), if one topology is finer or coarser than the other, then they must be equal (that is, if τ1⊆τ2\tau _{1}\subseteq \tau _{2} or τ2⊆τ1\tau _{2}\subseteq \tau _{1} then τ1=τ2\tau _{1}=\tau _{2} ). So, for example, if (X,p)(X,p) and (X,q)(X,q) are Banach spaces with topologies τp\tau _{p} and τq,\tau _{q}, and if one of these spaces has some open ball that is also an open subset of the other space (or, equivalently, if one of p:(X,τq)→Rp:(X,\tau _{q})\to \mathbb {R} or q:(X,τp)→Rq:(X,\tau _{p})\to \mathbb {R} is continuous), then their topologies are identical and the norms pp and qq are equivalent. Two norms, pp and q,q, on a vector space XX are said to be equivalent if they induce the same topology; this happens if and only if there exist real numbers c,C>0c,C>0 such that c q(x)≤p(x)≤C q(x){\textstyle c\,q(x)\leq p(x)\leq C\,q(x)} for all x∈X.x\in X. If pp and qq are two equivalent norms on a vector space XX then (X,p)(X,p) is a Banach space if and only if (X,q)(X,q) is a Banach space. See this footnote for an example of a continuous norm on a Banach space that is not equivalent to that Banach space's given norm. All norms on a finite-dimensional vector space are equivalent and every finite-dimensional normed space is a Banach space. A metric DD on a vector space XX is induced by a norm on XX if and only if DD is translation invariant and absolutely homogeneous, which means that D(sx,sy)=∣s∣D(x,y)D(sx,sy)=|s|D(x,y) for all scalars ss and all x,y∈X,x,y\in X, in which case the function ∥x∥:=D(x,0)\|x\|:=D(x,0) defines a norm on XX and the canonical metric induced by ∥⋅∥\|{\cdot }\| is equal to D.D. Suppose that (X,∥⋅∥)(X,\|{\cdot }\|) is a normed space and that τ\tau is the norm topology induced on X.X. Suppose that DD is anymetric on XX such that the topology that DD induces on XX is equal to τ.\tau . If DD is translation invariant then (X,∥⋅∥)(X,\|{\cdot }\|) is a Banach space if and only if (X,D)(X,D) is a complete metric space. If DD is not translation invariant, then it may be possible for (X,∥⋅∥)(X,\|{\cdot }\|) to be a Banach space but for (X,D)(X,D) to not be a complete metric space (see this footnote for an example). In contrast, a theorem of Klee, which also applies to all metrizable topological vector spaces, implies that if there exists any complete metric DD on XX that induces the norm topology τ\tau on X,X, then (X,∥⋅∥)(X,\|{\cdot }\|) is a Banach space. A Fréchet space is a locally convex topological vector space whose topology is induced by some translation-invariant complete metric. Every Banach space is a Fréchet space but not conversely; indeed, there even exist Fréchet spaces on which no norm is a continuous function (such as the space of real sequences RN=∏i∈NR{\textstyle \mathbb {R} ^{\mathbb {N} }=\prod _{i\in \mathbb {N} }\mathbb {R} } with the product topology). However, the topology of every Fréchet space is induced by some countable family of real-valued (necessarily continuous) maps called seminorms, which are generalizations of norms. It is even possible for a Fréchet space to have a topology that is induced by a countable family of norms (such norms would necessarily be continuous) but to not be a Banach/normable space because its topology can not be defined by any single norm. An example of such a space is the Fréchet space C∞(K),C^{\infty }(K), whose definition can be found in the article on spaces of test functions and distributions. There is another notion of completeness besides metric completeness and that is the notion of a complete topological vector space (TVS) or TVS-completeness, which uses the theory of uniform spaces. Specifically, the notion of TVS-completeness uses a unique translation-invariant uniformity, called the canonical uniformity, that depends only on vector subtraction and the topology τ\tau that the vector space is endowed with, and so in particular, this notion of TVS completeness is independent of whatever norm induced the topology τ\tau (and even applies to TVSs that are not even metrizable). Every Banach space is a complete TVS. Moreover, a normed space is a Banach space (that is, its norm-induced metric is complete) if and only if it is complete as a topological vector space. If (X,τ)(X,\tau ) is a metrizable topological vector space (such as any norm induced topology, for example), then (X,τ)(X,\tau ) is a complete TVS if and only if it is a sequentially complete TVS, meaning that it is enough to check that every Cauchy sequence in (X,τ)(X,\tau ) converges in (X,τ)(X,\tau ) to some point of XX (that is, there is no need to consider the more general notion of arbitrary Cauchy nets). If (X,τ)(X,\tau ) is a topological vector space whose topology is induced by some (possibly unknown) norm (such spaces are called normable), then (X,τ)(X,\tau ) is a complete topological vector space if and only if XX may be assigned a norm ∥⋅∥\|{\cdot }\| that induces on XX the topology τ\tau and also makes (X,∥⋅∥)(X,\|{\cdot }\|) into a Banach space. A Hausdorfflocally convex topological vector space XX is normable if and only if its strong dual space Xb′X'_{b} is normable, in which case Xb′X'_{b} is a Banach space ( Xb′X'_{b} denotes the strong dual space of X,X, whose topology is a generalization of the dual norm-induced topology on the continuous dual space X′X' ; see this footnote for more details). If XX is a metrizable locally convex TVS, then XX is normable if and only if Xb′X'_{b} is a Fréchet–Urysohn space. This shows that in the category of locally convex TVSs, Banach spaces are exactly those complete spaces that are both metrizable and have metrizable strong dual spaces. Every normed space can be isometrically embedded onto a dense vector subspace of a Banach space, where this Banach space is called a completion of the normed space. This Hausdorff completion is unique up to isometric isomorphism. More precisely, for every normed space X,X, there exists a Banach space YY and a mapping T:X→YT:X\to Y such that TT is an isometric mapping and T(X)T(X) is dense in Y.Y. If ZZ is another Banach space such that there is an isometric isomorphism from XX onto a dense subset of Z,Z, then ZZ is isometrically isomorphic to Y.Y. The Banach space YY is the Hausdorff completion of the normed space X.X. The underlying metric space for YY is the same as the metric completion of X,X, with the vector space operations extended from XX to Y.Y. The completion of XX is sometimes denoted by X^.{\widehat {X}}. Cite error: There are tags on this page, but the references will not show without a {{reflist|group=note}} template (see the help page). - ^ ^{a}^{b}Lumer, G. (1961). "Semi-inner-product spaces". Transactions of the American Mathematical Society. 100 (1): 29–43. doi:10.1090/S0002-9947-1961-0133024-2. - ^see Theorem 1.3.9, p. 20 in Megginson (1998). - ^Wilansky 2013, p. 29. - ^Bessaga & Pełczyński 1975, p. 189 - ^ ^{a}^{b}Anderson & Schori 1969, p. 315. - ^Henderson 1969. - ^Aliprantis & Border 2006, p. 185. - ^Trèves 2006, p. 145. - ^Trèves 2006, pp. 166–173. - ^ ^{a}^{b}Conrad, Keith. "Equivalence of norms"(PDF). kconrad.math.uconn.edu. Archived(PDF) from the original on 2022-10-09. Retrieved September 7, 2020. - ^see Corollary 1.4.18, p. 32 in Megginson (1998). - ^Narici & Beckenstein 2011, pp. 47–66. - ^Narici & Beckenstein 2011, pp. 47–51. - ^Schaefer & Wolff 1999, p. 35. - ^Klee, V. L. (1952). "Invariant metrics in groups (solution of a problem of Banach)"(PDF). Proc. Amer. Math. Soc. 3 (3): 484–487. doi:10.1090/s0002-9939-1952-0047250-4. Archived(PDF) from the original on 2022-10-09. - ^Trèves 2006, pp. 57–69. - ^Trèves 2006, p. 201. - ^Gabriyelyan, S.S. "On topological spaces and topological groups with certain local countable networks (2014)
Here are the main general results about Banach spaces that go back to the time of Banach's book (Banach (1932)) and are related to the Baire category theorem. According to this theorem, a complete metric space (such as a Banach space, a Fréchet space or an F-space) cannot be equal to a union of countably many closed subsets with empty interiors. Therefore, a Banach space cannot be the union of countably many closed subspaces, unless it is already equal to one of them; a Banach space with a countable Hamel basis is finite-dimensional. Banach–Steinhaus Theorem—Let XX be a Banach space and YY be a normed vector space. Suppose that FF is a collection of continuous linear operators from XX to Y.Y. The uniform boundedness principle states that if for all xx in XX we have sup⁡T∈F∥T(x)∥Y<∞,\sup _{T\in F}\|T(x)\|_{Y}<\infty , then sup⁡T∈F∥T∥Y<∞.\sup _{T\in F}\|T\|_{Y}<\infty . The Banach–Steinhaus theorem is not limited to Banach spaces. It can be extended for example to the case where XX is a Fréchet space, provided the conclusion is modified as follows: under the same hypothesis, there exists a neighborhood UU of 0\mathbf {0} in XX such that all TT in FF are uniformly bounded on U,U, sup⁡T∈Fsup⁡x∈U  ∥T(x)∥Y<∞.\sup _{T\in F}\sup _{x\in U}\;\|T(x)\|_{Y}<\infty . The Open Mapping Theorem—Let XX and YY be Banach spaces and T:X→YT:X\to Y be a surjective continuous linear operator, then TT is an open map. Corollary—Every one-to-one bounded linear operator from a Banach space onto a Banach space is an isomorphism. The First Isomorphism Theorem for Banach spaces—Suppose that XX and YY are Banach spaces and that T∈B(X,Y).T\in B(X,Y). Suppose further that the range of TT is closed in Y.Y. Then X/ker⁡TX/\ker T is isomorphic to T(X).T(X). This result is a direct consequence of the preceding Banach isomorphism theorem and of the canonical factorization of bounded linear maps. Corollary—If a Banach space XX is the internal direct sum of closed subspaces M1,…,Mn,M_{1},\ldots ,M_{n}, then XX is isomorphic to M1⊕⋯⊕Mn.M_{1}\oplus \cdots \oplus M_{n}. This is another consequence of Banach's isomorphism theorem, applied to the continuous bijection from M1⊕⋯⊕MnM_{1}\oplus \cdots \oplus M_{n} onto XX sending m1,⋯ ,mnm_{1},\cdots ,m_{n} to the sum m1+⋯+mn.m_{1}+\cdots +m_{n}. The Closed Graph Theorem—Let T:X→YT:X\to Y be a linear mapping between Banach spaces. The graph of TT is closed in X×YX\times Y if and only if TT is continuous.
Let K\mathbb{K} be the field of scalars, R\mathbb{R} or C\mathbb{C}. A Banach space is a pair (V,∣∣⋅∣∣)(V, ||\cdot||), where VV is a vector space over K\mathbb{K}, and ∣⋅∣∣|\cdot|| is a norm on VV, such that the induced metric d(x,y)=∣∣x−y∣∣d(x, y) = ||x - y|| makes VV a complete metric space. \n\nFormally, the norm ∣⋅∣∣:V→R|\cdot||: V \to \mathbb{R} must satisfy the following axioms for all x,y∈Vx, y \in V and α∈K\alpha \in \mathbb{K}:\n\n1. Non-negativity and Definiteness: ∣∣x∣∣≥0||x|| \ge 0, and ∣∣x∣∣=0  ⟺  x=0||x|| = 0 \iff x = 0.\n2. Homogeneity: ∣αx∣∣=∣α∣⋅∣∣x∣∣|\alpha x|| = |\alpha| \cdot ||x||.\n3. Triangle Inequality: ∣∣x+y∣∣≤∣∣x∣∣+∣∣y∣∣||x + y|| \le ||x|| + ||y||.\n\nFurthermore, VV must be complete, meaning that every Cauchy sequence in VV converges to an element in VV. That is, for any sequence (xn)n=1∞⊂V(x_n)_{n=1}^{\infty} \subset V, if for every ϵ>0\epsilon > 0, there exists an integer N∈NN \in \mathbb{N} such that for all m,n≥Nm, n \ge N, we have ∣∣xm−xn∣∣<ϵ||x_m - x_n|| < \epsilon, then there exists an element x∈Vx \in V such that lim⁡n→∞∣∣xn−x∣∣=0\lim_{n \to \infty} ||x_n - x|| = 0.
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Allows the extension of bounded linear functionals defined on a subspace to the whole space while preserving the norm.