Banach space
B-space
2010 Mathematics Subject Classification: Primary: 46B Secondary: 46E15 [MSN][ZBL] $ \newcommand{\abs}[1]{\left|#1\right|} \newcommand{\norm}[1]{\left\|#1\right\|} \newcommand{\set}[1]{\left\{#1\right\}} $
A complete normed vector space. The problems involved in Banach spaces are of different types: the geometry of the unit ball, the geometry of subspaces, the linear topological classification, series and sequences in Banach spaces, best approximations in Banach spaces, functions with values in a Banach space, etc. Regarding the theory of operators in Banach spaces it should be pointed out that many theorems are directly related to the geometry and the topology of Banach spaces.
Contents
History
The function spaces introduced by D. Hilbert, M. Fréchet and F. Riesz between 1904 and 1918 served as the starting point for the theory of Banach spaces. It is in these spaces that the fundamental concepts of strong and weak convergence, compactness, linear functional, linear operator, etc., were originally studied. Banach spaces were named after S. Banach who in 1922 began a systematic study of these spaces [Ba], based on axioms introduced by himself, and who obtained highly advanced results.
The theory of Banach spaces developed in parallel with the general theory of linear topological spaces. These theories mutually enriched one another with new ideas and facts. Thus, the idea of semi-norms, taken from the theory of normed spaces, became an indispensable tool in constructing the theory of locally convex linear topological spaces. The ideas of weak convergence of elements and linear functionals in Banach spaces ultimately evolved to the concept of weak topology. The theory of Banach spaces is a thoroughly studied branch of functional analysis, with numerous applications in various branches of mathematics — directly or by way of the theory of operators.
Generalities
A Banach space $X$ is a vector space over $\R$ or $\C$ with a norm $\norm{\cdot}$ which is complete with respect to this norm, i.e., every Cauchy sequence in $X$ converges.
For two Banach spaces $X$, $Y$, denote by $B(X,Y)$ the space of linear continuous maps from $X$ to $Y$. It is in itself a Banach space with respect to the norm $$\norm{T} = \sup_{x \neq 0} \frac{\norm{Tx}}{\norm{x}}.$$
Examples
The Banach spaces encountered in analysis are mostly sets of functions or sequences of numbers which are subject to certain conditions.
- $\ell_p$, $p \geq 1$, is the space of numerical sequences $\set{\xi_n}$ for which $$ \sum_{n=1}^\infty \abs{\xi_n}^p < \infty$$ with the norm $$ \norm{x} = \left( \sum_{n=1}^\infty \abs{\xi_n}^p \right)^{1/p}. $$
- $m$ is the space of bounded numerical sequences with the norm $$ \norm{x} = \sup_n\abs{\xi_n}.$$
- $c$ is the space of convergent numerical sequences with the norm $$\norm{x} = \sup_n\abs{\xi_n}.$$
- $c_0$ is the space of numerical sequences which converge to zero with the norm $$ \norm{x} = \max_n\abs{\xi_n}.$$
- $C[a,b]$ is the space of continuous functions $x=x(t)$ on $[a,b]$ with the norm $$\norm{x} = \max_{a \leq t \leq b}\abs{x(t)}.$$
- $C[K]$ is the space of continuous functions on a compactum $K$ with the norm $$\norm{x} = \max_{t \in K}\abs{x(t)}$$.
- $C^n[a,b]$ is the space of functions with continuous derivatives up to and including the order $n$, with the norm $$\norm{x} = \sum_{k=0}^n \max_{a \leq t \leq b}\abs{x^{(k)}(t)}. $$
- $C^n[I^m]$ is the space of all functions defined in an $m$-dimensional cube that are continuously differentiable up to and including the order $n$, with the norm of uniform boundedness in all derivatives of order at most $n$. (Cf. Hölder space.)
- $M[a,b]$ is the space of bounded measurable functions with the norm $$\norm{x} = \mathop{\mathrm{ess\;max}}_{a \leq t \leq b} \abs{x(t)}.$$
- $A(D)$ is the space of functions which are analytic in the open unit disc $D$ and are continuous in the closed disc $\bar{D}$, with the norm $$\norm{x} = \max_{z \in \bar{D}}\abs{x(z)}. $$
- $L_p(S ; \Sigma, \mu)$, $p \geq 1$, is the space of functions $x(s)$ defined on a set $S$ provided with a countably-additive measure $\mu$, with the norm $$\norm{x} = \left( \int_S \abs{x(s)}^p \,\mu(\mathrm{d}s) \right)^{1/p}.$$ (Cf. $L^p$ spaces.)
- $L_p[a,b]$, $p \geq 1$, is a special case of the space $L_p(S ; \Sigma, \mu)$. It is the space of Lebesgue-measurable functions, summable of degree $p$, with the norm $$\norm{x} = \left( \int_a^b \abs{x(s)}^p \,\mathrm{d}s \right)^{1/p}.$$
- $AP$ is the Bohr space of almost-periodic functions, with the norm $$\norm{x} = \sup_{-\infty < t < \infty} \abs{x(t)}. $$
The spaces $C[a,b]$, $C^n[a,b]$, $L_p[a,b]$, $c$, $\ell_p$ are separable; the spaces $M[a,b]$, $m$, $AP$ are non-separable; $C[K]$ is separable if and only if $K$ is a compact metric space.
Other examples include Sobolev spaces and the Hardy space $\mathcal{H}^1$. All Hilbert spaces are a forteriori Banach spaces.
Quotients
A (closed linear) subspace $Y$ of a Banach space, considered apart from the enveloping space $X$, is a Banach space. The quotient space $X/Y$ of a normed space by a subspace $Y$ is a normed space if the norm is defined as follows. Let $Y_1 = x_1 + Y$ be a coset. Then $$ \norm{Y_1} = \inf_{y \in Y} \norm{x_1 + y}. $$ If $X$ is a Banach space, then $X/Y$ is a Banach space as well.
In this case, if $Z$ is another normed space and $T\in B(X,Z)$ fulfills $T(Y)=\{0\}$, then there exists $\hat T \in B(X/Y,Z)$ such that $T = \hat T \circ Q$ and $\norm{T}=\norm{\hat T}$, where $Q:X \to X/Y$ is the quotient mapping. [KR1, Theorem 1.5.8]
Linear functionals, dual space
The set of all continuous linear functionals defined on the normed space $X$, with the norm $$ \norm{f} = \sup_{x \in X} \frac{\abs{f(x)}}{\norm{x}}, \quad x \neq 0 $$ is said to be the dual space of $X$, and is denoted by $X^*$. It is a Banach space.
Hahn-Banach theorem
Banach spaces satisfy the Hahn–Banach theorem on the extension of linear functionals: If a linear functional is defined on a subspace $Y$ of a normed space $X$, it can be extended, while preserving its linearity and continuity, onto the whole space $X$. Moreover, the extension can be made to have the same norm: $$ \norm{f}_X = \sup_{x \in X} \frac{\abs{f(x)}}{\norm{x}} = \norm{f}_Y = \sup_{y \in Y} \frac{\abs{f(y)}}{\norm{y}}. $$ Even a more general theorem is valid: Let a real-valued function $p(x)$ defined on a linear space satisfy the conditions: $$ p(x+y) \leq p(x) + p(y), \quad p(\lambda x) = \lambda p(x), \quad \lambda \geq 0, \quad x,y \in X, $$ and let $f(x)$ be a real-valued linear functional defined on a subspace $Y \subset X$ and such that $$ f(x) \leq p(x), \quad x \in Y. $$ Then there exists a linear functional $F(x)$ defined on the whole of $X$ such that $$ F(x) = f(x), \quad x \in Y; \quad F(x) \leq p(x), \quad x \in X. $$ A consequence of the Hahn–Banach theorem is the "inverse" formula which relates the norms of $X$ and $X^*$: $$ \norm{x} = \max_{f \in X^*} \frac{\abs{f(x)}}{\norm{f}},\quad f \neq 0, \quad x \in X. $$ The maximum in this formula is attained for some $f=f_X\in X^*$. Another important consequence is the existence of a separating set of continuous linear functionals, meaning that for any $x_1 \neq x_2 \in X$ there exists a linear functional $f$ on $X$ such that $f(x_1) \neq f(x_2)$ (cf. Complete set of functionals).
General structure of linear functionals
The general form of a linear functional is known for many specific Banach spaces. Thus, on $L_p[a,b]$, $p>1$, all linear functionals are given by a formula $$ f(x) = \int_a^b x(t)y(t) \,\mathrm{d}t, $$ where $y \in L_q[a,b]$, $1/p + 1/q = 1$, and any function $y(t) \in L_q$ defines a linear functional $f$ by this formula, moreover $$ \norm{f} = \left( \int_a^b \abs{y(t)}^q \,\mathrm{d}t \right)^{1/q}. $$ Thus, the dual space of $L_p$ is $L_q$: $L_p^* = L_q$. Linear functionals on $L_1[a,b]$ are defined by the same formula, but in this case $y \in M$, so that $L_1^* = M$.
Biduals, reflexivity
The space $X^{**}$, dual to $X^*$, is said to be the second dual or bidual. Third, fourth, etc., dual spaces are defined in a similar manner. Each element in $X$ may be identified with some linear functional defined on $X^*$: $$ \text{$F(f) = f(x)$ for all $f \in X^*$ ($F \in X^{**}$, $x \in X$),} $$ where $\norm{F} = \norm{x}$. One may then regard $X$ as a subspace of the space $X^{**}$ and $X \subset X^{**} \subset X^\text{IV} \subset \cdots$, $X^* \subset X^{***} \subset \cdots$. If, as a result of these inclusions, the Banach space coincides with its second dual, it is called reflexive. In such a case all inclusions are equalities. If $X$ is not reflexive, all inclusions are strict. If the quotient space $X^{**}/X$ has finite dimension $n$, $X$ is said to be quasi-reflexive of order $n$. Quasi-reflexive spaces exist for all $n$.
- Reflexivity criteria for Banach spaces
- $X$ is reflexive if and only if for each $f \in X^*$ it is possible to find an $x \in X$ on which the "sup" in the formula $$ \norm{f} = \sup_{x \in X} \frac{\abs{f(x)}}{\norm{x}}, \quad x \neq 0, $$ is attained.
- In reflexive Banach spaces and only in such spaces each bounded set is relatively compact with respect to weak convergence: Any one of its infinite parts contains a weakly convergent sequence (the Eberlein–Shmul'yan theorem). The spaces $L_p$ and $\ell_p$, $p>1$, are reflexive. The spaces $L_1$, $\ell_1$, $C$, $M$, $c$, $m$, $AP$ are non-reflexive.
Special cases
Weakly complete spaces
A Banach space is said to be weakly complete if each weak Cauchy sequence in it weakly converges to an element of the space. Every reflexive space is weakly complete. Moreover, the Banach spaces $L_1$ and $\ell_1$ are weakly complete. The Banach spaces not containing a subspace isomorphic to $c_0$ form an even wider class. These spaces resemble weakly-complete spaces in several respects.
Strictly convex spaces
A Banach space is said to be strictly convex if its unit sphere $S$ contains no segments. Convexity moduli are introduced for a quantitative estimation of the convexity of the unit sphere; these are the local convexity modulus $$ \delta(x,\epsilon) = \inf\set{ 1 - \norm{\frac{x+y}{2}} : y \in S,\, \norm{x-y} \geq \epsilon}, \quad x \in S, \quad 0 < \epsilon \leq 2, $$ and the uniform convexity modulus $$ \delta(\epsilon) = \inf_{x \in S} \delta(x,\epsilon). $$ If $\delta(x,\epsilon) > 0$ for all $x \in S$ and all $\epsilon > 0$, the Banach space is said to be locally uniformly convex. If $\delta(x) > 0$, the space is said to be uniformly convex. All uniformly convex Banach spaces are locally uniformly convex; all locally uniformly convex Banach spaces are strictly convex. In finite-dimensional Banach spaces the converses are also true. If a Banach space is uniformly convex, it is reflexive.
Smooth spaces
A Banach space is said to be smooth if for any linearly independent elements $x$ and $y$ the function $\psi(t)=\norm{x+ty}$ is differentiable for all values of $t$. A Banach space is said to be uniformly smooth if its modulus of smoothness $$ \rho(t) = \sup_{x,y \in S} \set{\frac{\norm{x + \tau y} + \norm{x - \tau y}}{2} -1}, \quad \tau > 0, $$ satisfies the condition $$ \lim_{\tau \rightarrow 0}\frac{\rho(\tau)}{\tau} = 0. $$ In uniformly smooth spaces, and only in such spaces, the norm is uniformly Fréchet differentiable. A uniformly smooth Banach space is smooth. The converse is true if the Banach space is finite-dimensional. A Banach space $X$ is uniformly convex (uniformly smooth) if and only if $X^*$ is uniformly smooth (uniformly convex). The following relationship relates the convexity modulus of a Banach space $X$ and the smoothness modulus of $X^*$: $$ \rho_{X^*}(\tau) = \sup_{0 < \epsilon \leq 2} \set{\frac{\epsilon\tau}{2} - \delta_X(\epsilon)}. $$ If a Banach space is uniformly convex (uniformly smooth), so are all its subspaces and quotient spaces. The Banach spaces $L_p$ and $\ell_p$, $p>1$, are uniformly convex and uniformly smooth, and $$ \delta(\epsilon) \simeq \begin{cases} \epsilon^2 & (1 < p \leq 2) \\ \epsilon^p & (2 \leq p < \infty); \end{cases} $$ $$ \rho(\tau) \simeq \begin{cases} \tau^p & (1 < p \leq 2) \\ \tau^2 & (2 \leq p < \infty); \end{cases} $$ $$ \left( f(\epsilon) \simeq \phi(\epsilon) \Leftrightarrow a < \frac{f(\epsilon)}{\phi(\epsilon)} < b \right). $$ The Banach spaces $M$, $C$, $A$, $L_1$, $AP$, $m$, $c$, $\ell_1$ are not strictly convex and are not smooth.
Linear operators
The following important theorems for linear operators are valid in Banach spaces:
If a family of linear operators $T=\set{T_\alpha}$ is bounded at each point, $$ \sup_\alpha \norm{T_\alpha x} < \infty, \quad x \in X, $$ then it is norm-bounded: $$ \sup_\alpha \norm{T_\alpha} < \infty. $$
- The Banach open-mapping theorem.
If a linear continuous operator maps a Banach space $X$ onto a Banach space $Y$ in a one-to-one correspondence, the inverse operator $T^{-1}$ is also continuous.
- The closed-graph theorem.
If a closed linear operator maps a Banach space $X$ into a Banach space $Y$, then it is continuous.
Isometries and isomorphisms
Isometries between Banach spaces occur rarely. The classical example is given by the Banach spaces $L_1$ and $\ell_2$. The Banach spaces $C[K_1]$ and $C[K_2]$ are isometric if and only if $K_1$ and $K_2$ are homeomorphic (the Banach–Stone theorem). A measure of proximity of isomorphic Banach spaces is the number $$ d(X,Y) = \ln\inf\bigl\|T\bigr\|\bigl\|T^{-1}\bigr\|, $$ where $T$ runs through all possible operators which realize a (linear topological) isomorphism between $X$ and $Y$. If $X$ is isometric to $Y$, then $d(X,Y)=0$. However, non-isometric spaces for which $d(X,Y)=0$ also exist; they are said to be almost-isometric. The properties of Banach spaces preserved under an isomorphism are said to be linear topological. They include separability, reflexivity and weak completeness. The isomorphic classification of Banach spaces contains, in particular, the following theorems: $$ L_r \neq L_s; \quad \ell_r \neq \ell_s, \quad r \neq s $$ $$ L_r \neq \ell_s, \quad r \neq s; \quad L_r = \ell_s, \quad r = s = 2; $$ $$ M=m; \quad C[0,1] \neq A(D); $$ $C[K] = C[0,1]$ if $K$ is a metric compactum with the cardinality of the continuum; $$ C^n[I^m] \neq C[0,1]. $$
Each separable Banach space is isomorphic to a locally uniformly convex Banach space. It is not known (1985) if there are Banach spaces which are isomorphic to none of their hyperplanes. There exist Banach spaces which are not isomorphic to strictly convex spaces. Irrespective of the linear nature of normed spaces, it is possible to consider their topological classification. Two spaces are homeomorphic if a one-to-one continuous correspondence, such that its inverse is also continuous, can be established between their elements. An incomplete normed space is not homeomorphic to any Banach space. All infinite-dimensional separable Banach spaces are homeomorphic.
In the class of separable Banach spaces, $C[0,1]$ and $A(D)$ are universal (cf. Universal space). The class of reflexive separable Banach spaces contains even no isomorphic universal spaces. The Banach space $\ell_1$ is universal in a somewhat different sense: All separable Banach spaces are isometric to one of its quotient spaces.
Non-complementable subspaces
Each of the Banach spaces mentioned above, except $L_2$ and $\ell_2$, contains subspaces without a complement. In particular, in $m$ and $M$ every infinite-dimensional separable subspace is non-complementable, while in $C[0,1]$ all infinite-dimensional reflexive subspaces are non-complementable. If all subspaces in a Banach space are complementable, the space is isomorphic to a Hilbert space. It is not known (1985) whether or not all Banach spaces are direct sums of some two infinite-dimensional subspaces. A subspace $Y$ is complementable if and only if there exists a projection which maps $X$ onto $Y$. The lower bound of the norms of the projections on $Y$ is called the relative projection constant $\lambda(Y,X) $ of the subspace $Y$ in $X$. Each $n$-dimensional subspace of a Banach space is complementable and $\lambda(Y_n,X) \leq \sqrt{n}$. The absolute projection constant $\lambda(Y)$ of a Banach space $Y$ is $$ \lambda(Y) = \sup_X \lambda(Y,X), $$ where $X$ runs through all Banach spaces which contain $Y$ as a subspace. For any infinite-dimensional separable Banach space $Y$ one has $\lambda(Y) = \infty$. Banach spaces for which $\lambda(Y) \leq Y < \infty$ form the class $\mathcal{P}_\lambda$ ($\lambda \geq 1$). The class $\mathcal{P}_1$ coincides with the class of spaces $C(Q)$ where $Q$ are extremally-disconnected compacta (cf. Extremally-disconnected space).
Finite-dimensional case
Fundamental theorems on finite-dimensional Banach spaces:
- A finite-dimensional space is complete, i.e. is a Banach space.
- All linear operators in a finite-dimensional Banach space are continuous.
- A finite-dimensional Banach space is reflexive (the dimension of $X^*$ is equal to the dimension of $X$).
- A Banach space is finite-dimensional if and only if its unit ball is compact.
- All $n$-dimensional Banach spaces are pairwise isomorphic; their set becomes compact if one introduces the distance
$$ d(X,Y) = \ln\inf_T\bigl\|T\bigr\|\bigl\|T^{-1}\bigr\|. $$
Convergence of series
A series \begin{equation} \sum_{k=1}^\infty x_k, \quad x_k \in X \label{eq:series} \end{equation} is said to be convergent if there exists a limit $S$ of the sequence of partial sums: $$ \lim_{n \rightarrow \infty} \norm{S - \sum_{k=1}^n x_k} = 0. $$ If $$ \sum_{k=1}^\infty \norm{x_k} < \infty, $$ the series $\eqref{eq:series}$ is convergent, and is said in such a case to be absolutely convergent. A series is said to be unconditionally convergent if it converges when its terms are arbitrarily rearranged. The sum of an absolutely convergent series is independent of the arrangement of its terms. In the case of series in a finite-dimensional space (and, in particular, for series of numbers) unconditional and absolute convergence are equivalent. In infinite-dimensional Banach spaces unconditional convergence follows from absolute convergence but the converse is not true in any infinite-dimensional Banach space. This is a consequence of the Dvoretskii–Rogers theorem: For all numbers $\alpha_k \geq 0$, subject to the condition $\sum\alpha_k^2 < \infty$, there exists in each infinite-dimensional Banach space an unconditionally convergent series $\sum x_k$ such that $\norm{x_k} = \alpha_k$, $k=1,2,\ldots$. In the space $c_0$ (and hence also in any Banach space containing a subspace isomorphic to $c_0$), for any sequence $\alpha_k \geq 0$ that converges to zero, there exists an unconditionally convergent series $\sum x_k$, $\norm{x_k} = \alpha_k$. In $L_p(S ; \Sigma, \mu)$ the unconditional convergence of the series $\sum x_k$ implies that $$ \sum_{k=1}^\infty \norm{x_k}^s < \infty, $$ where $$ s = \begin{cases} 2 & (1 \leq p \leq 2), \\ p & (p \geq 2). \end{cases} $$ In a uniformly convex Banach space with convexity modulus $\delta(\epsilon)$ the unconditional convergence of the series $\sum x_k$ implies that $$ \sum_{k=1}^\infty\delta(\norm{x_k}) < \infty. $$
A series $\sum x_k$ is said to be weakly unconditionally Cauchy if the series of numbers $\sum\abs{f(x_k)}$ converges for each $f \in X^*$. Each weakly unconditionally Cauchy series in $X$ converges if and only if $X$ contains no subspace isomorphic to $c_0$.
A sequence of elements $\set{e_k}_1^\infty$ of a Banach space is said to be minimal if each one of its terms lies outside the closure of $X^{(n)} = [e_k]_{k \neq n}$, the linear hull of the remaining elements. A sequence is said to be uniformly minimal if $$ \rho(e_n ; X^{(n)}) \geq \gamma\norm{e_n}, \quad 0 < \gamma \leq 1, \quad n = 1, 2, \ldots. $$ If $\gamma=1$, the series is said to be an Auerbach system. In each $n$-dimensional Banach space there exists a complete Auerbach system $\set{e_k}_1^n$. It is not known (1985) whether or not a complete Auerbach system exists in each separable Banach space. For each minimal system there exists an adjoint system of linear functionals $\set{f_n}$, which is connected with $\set{e_k}$ by the biorthogonality relations: $f_i(e_j) = \delta_{ij}$. In such a case the system $\set{e_k,f_k}$ is said to be biorthogonal. A set of linear functionals is said to be total if it annihilates only the zero element of the space. In each separable Banach space there exists a complete, minimal system with a total adjoint. Each element $x \in X$ can formally be developed in a series by the biorthogonal system: $$ x \sim \sum_{k=1}^\infty f_k(x)e_k, $$ but in the general case this series is divergent.
Bases
A system of elements $\set{e_k}_1^\infty$ is said to be a basis in $X$ if each element $x \in X$ can be uniquely represented as a convergent series $$ x = \sum_{k=1}^\infty \alpha_k e_k, \quad \alpha_k = \alpha_k(x). $$ Each basis in a Banach space is a complete uniform minimal system with a total adjoint. The converse is not true, as can be seen from the example of the system $\set{e^{int}}_{-\infty}^\infty$ in $C[0,2\pi]$ and $L_1[0,2\pi]$.
A basis is said to be unconditional if all its rearrangements are also bases; otherwise it is said to be conditional. The system $\set{e^{int}}_{-\infty}^\infty$ in $L_p[0,2\pi]$, $p>1$, $p \neq 2$, is a conditional basis. The Haar system is an unconditional basis in $L_p$, $p > 1$. There is no unconditional basis in the spaces $C$ and $L_1$. It is not known (1985) whether or not each Banach space contains an infinite-dimensional subspace with an unconditional basis. Any non-reflexive Banach space with an unconditional basis contains a subspace isomorphic to $\ell_1$ or $c_0$.
Two normalized bases $\set{e_k^\prime}$ and $\set{e_k^{\prime\prime}} $ in two Banach spaces $X_1$ and $X_2$ are said to be equivalent if the correspondence $e_k^\prime \leftrightarrow e_k^{\prime\prime}$, $k=1,2,\ldots$, may be extended to an isomorphism between $X_1$ and $X_2$. In each of the spaces $\ell_2$, $\ell_1$, $c_0 $ all normalized unconditional bases are equivalent to the natural basis. Bases constructed in Banach spaces which have important applications are not always suitable for solving problems, e.g. in the theory of operators. $T$-bases, or summation bases, have been introduced in this context. Let $\set{t_{i,j}}_1^\infty$ be the matrix of a regular summation method. The system of elements $\set{e_n} \subset X$ is said to be a $T$-basis corresponding to the given summation method if each $x \in X$ can be uniquely represented by a series $$ x \sim \sum_{k=1}^\infty \alpha_k e_k, $$ which is summable to $x$ by this method. The trigonometric system $\set{e^{int}}_{-\infty}^\infty$ in $C[0,2\pi]$ is a summation basis for the methods of Cesàro and Abel. Each $T$-basis is a complete minimal (not necessarily uniformly minimal) system with a total adjoint. The converse is not true. Until the 1970s, one of the principal problems of the theory of Banach spaces was the basis problem dealt with by Banach himself: Does a basis exist in each separable Banach space? The question of existence of a basis in specifically defined Banach spaces remained open as well. The first example of a separable Banach space without a basis was constructed in 1972; bases in the spaces $C^n(I^m)$ and $A(D)$ have been constructed.
References
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Banach space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Banach_space&oldid=43786