A Banach space is a complete normed vector space: A vector space equipped with a norm such that every Cauchy sequence converges.

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Matrix norm on $L^p$ operator algebras

I have a question about the framework of $L^p$ operator algebras ($1<p<\infty$), i.e. norm-closed subalgebras of $B(L^p(X,\mu))$ for some measure space $(X,\mu)$ (see e.g. ...
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Characterizations of Wiener algebra

The Wiener algebra $\mathcal W$ is defined as $\text{Fourier}(L^1(\mathbb R))$, i.e. the image by the Fourier transform of $L^1(\mathbb R)$. Riemann-Lebesgue's lemma ensures that $$ \mathcal W\subset ...
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Location of a Banach Space inside its bidual

Let $X$ be a Banach Space and let $Y$ be a closed subspace of $X^{**}$ such that $X\bigcap Y=0$. Let $P$ be the quotient map from $X^{**}$ onto $X^{**}/ Y$. I need to prove or refute that ...
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Are “most” operators on an infinite-dimensional complex Banach space “diagonalizable”?

This is true for finite-dimensional spaces: the diagonal operators on a finite dimensional complex vector space form contain a dense open set and the nondiagonalizable operators have measure 0. To be ...
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Estimates of entropy of functional spaces

Let $M^n$ be a compact $n$-dimensional manifold. For $k\geq 0$ let us denote by $C^k(M)$ the Banach space of $k$ times continuously differentiable functions, and $B_{C^k}$ denote the unit ball of it. ...
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Basis equivalent with a monotone basis

Given a basis in a Banach space $X$, can one find, for every $\varepsilon>0$, an equivalent basis with basis constant at most $1+\varepsilon$? In $L_p[0,1]$ with $1<p<\infty$ any monotone ...
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Between Tietze's and Dugundji's Extension Theorems

The celebrated Tietze Extension Theorem asserts that any continuous real-valued function defined on a closed subset of a normal space, can be extended to a continuous function on the whole space. Seen ...
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reflexive banach space

I want to ask this non-expert question: What does it mean geometrically for a Banach space to be reflexive? Well, we could say a Banach space is reflexive iff unit ball is weakly compact. Or some ...
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185 views

A question in Banach space

Given $A\in\Bbb R^{m\times n}$, define $$\gamma_2(A)=\min_{XY=A}\max_{i,j}\|x_i\|_{\ell_2}\|y_j\|_{\ell_2}$$ where rows of $X$,columns of $Y$ given by $\{x_i\}_{i=1}^m,\{y_j\}_{j=1}^n$ respectively. ...
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Lipschitz-free spaces of $\mathbb R^n$

We define $$ \text{Lip}_0(\mathbb R^n)=\{f:\mathbb R^n\rightarrow \mathbb R, \text{such that $f(0)=0$ and } \sup_{x\not=y}\frac{\vert f(x)-f(y)\vert}{\vert x-y\vert}<+\infty. \} $$ It is well-known ...
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Infinite direct sum of $l_2^{(n_k)}$ contains a complemented isometric copy of $l_2$

How do I show that for any increasing sequence $(n_k) \subseteq \mathbb{N}$, the space $\left( \oplus _{k=1} ^\infty l_2 ^{(n_k)} \right) _\infty$ contains a complemented isometric copy of $l_2$?
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Banach space dual to $L^\infty(I,H^1(M))$

What is the dual to $L^\infty (I,H^1(M))$?, where $I$ is an interval in the real line; $H^1(M)$ is Sobolev space of degree 1, and $M$ is a compact manifold like the torus. Any references that show ...
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1answer
111 views

Predual of a subspace

Let $E$ be a Banach space, let $d\ge 1$ be an integer. Let $\mathcal G$ be a weakly closed subspace of $(E^*)^d$ with finite codimension. I would like know if the space $\mathcal G$ is a dual space ...
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Inductive/Projective Limits of Topological Algebras

It is common to form inductive/projective limits of Banach/Frechet spaces in order to come up with natural topologies for common vector spaces. For instance, For $k \ge 0$ and $K_n$ compact ...
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Ultraweak topology on B(X): Is the map X\otimes X* -> B(X)* isometric?

Let $X$ be a Banach space. Consider the map $$ \alpha\colon X\hat{\otimes} X^* \to B(X)^*, $$ defined one simple tensors as $$ \alpha(\xi\otimes\eta)(a) = \eta(a(\xi)).\quad (\xi\in X, \eta\in X^*, ...
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Is this structure a Banach bundle?

Let $X$ be a Banach space. Put $Y=\{ \phi\in X^{*}\mid\;\; \parallel \phi \parallel\leq 1\;\; \&\;\; \phi \neq 0\}$ which is a locally compact Hausdorf space with the weak star topology. ...
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A question on metric characterization of approximation property

My question is: a Banach space $X$ has the approximation property if and only if for every $\epsilon>0$ and every compact subset $K\subset X$, there exists a Lipschitz map $T: X\rightarrow X$ with ...
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599 views

Non-separable Banach space

The vector space $C_b(\mathbb R)$ of bounded continuous functions on $\mathbb R$ is non-separable: it is possible to produce a direct proof of this fact, mimicking the standard proof for the ...
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Are $L^\infty(\Bbb R)$ and $L^2(\Bbb R)$ homeomorphic?

It's easy to see that, for $1\le p,q< \infty$ the spaces $L^p(\Bbb R)$ and $L^q(\Bbb R)$ of $p$-th and $q$-th power integrable functions on the real line are homeomorphic as topological spaces. In ...
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107 views

Approximation Property: Decomposition

This thread originated from MSE: Approximation Property: Decomposition Given a Banach space $E$. Consider a finite rank operator $F\in\mathcal{F}(X,E)$. Introduce a basis on the finite dimensional ...
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Approximation Property: Characterization

Problem Given a Banach space $E$. Denote compact sets by $\mathcal{C}$, compact operators by $\mathcal{C}(X,Y)$, and finite rank operators by $\mathcal{F}(X,Y)$. Suppose it has the approximation ...
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Compact Approximation

This thread originated from MSE: Compact Approximation This is meant as lemma for: Approximation Property Given a Banach space $E$. Denote compact domains by $\mathcal{C}$. Denote compact ...
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Non-reflexive Banach space s.t. X,X*,X**,… are separable

Is there an infinite-dimensional Banach space $X$, which is not reflexive, such that all the spaces $X,X^{\ast},X^{\ast\ast}, X^{\ast\ast\ast},\dots$ are separable?
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Can we conclude that $\varphi(L_xf_0)\neq0$ for every $x\in G$?

Let $H$ be a compact subgroup of locally compact topological group $G$ and $A=\{f\in L^1(G); R_hf=f(a,e)\forall h\in H\}$ as a subalgebra of $L^1(G)$ by convolution of $L^1(G)$. If $\varphi \in ...
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Dual of Banach-valued $L^p$ [duplicate]

Let $X$ be an infinite-dimensional Banach space and let $p\in(1,+\infty)$. We may define $L^p(\mathbb R;X)$. Is it always true that the topological dual of $L^p(\mathbb R;X)$ is $L^{p'}(\mathbb ...
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Completeness of spaces $\Lambda(\varphi, p)$

Definition 1. Let $f$ be measurable function on a mesurable subset $E\subset \mathbb R^n$. Non-increasing rearrangement of $f$ is a function $f^\ast(x)=\inf\{s>0: \operatorname{mes} E[|f| > s] ...
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103 views

Approximate singular value decomposition in Banach spaces

I am interested in generalisations to Banach spaces of the following construction, which relates to the singular value decomposition of a finite-dimensional linear map. If $V$, $W$ are ...
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A reasonable framework to study properties of operator $A \mapsto KAK$ on Banach space

Let $K$ be a continuous linear operator on $C[0,1]$ (more, precisely, it is a linear integral operator). Then $K$ defines a continous linear operator $\widehat K$ on $\mathcal L(C[0,1])$ by the rule ...
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Integration in C^* algebra

Let $\mathfrak{A}$ be a C${}^*$ algebra and $\mathbb{R}\ni s \mapsto \alpha_s$ a continuous family of its automorphisms. Is it true that $$ \int d s \, f(s)\, \alpha_s(A) $$ is well defined as a ...
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On a variant of Eidelheit's theorem

A theorem of Eidelheit from 1940's asserts that two Banach spaces $X$ and $Y$ are isomorphic if and only if $L(X)$ and $L(Y)$, the algebras of all bounded linear operators, are isomorphic as Banach ...
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Can sum of two (co)type $p$ subspaces fail to have the same (co)type?

Type and cotype are very pleasant invariants of Banach spaces. However answer to the following question seems to be missing from the literature. Let $X$ be a Banach space and suppose that $Y,Z$ are ...
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ODE properties true in finite dimension but not in Banach spaces of infinite dimension

Some properties of Ordinary Differential Equations - ODE are true in finite dimension spaces but not in Banach spaces of infinite dimension. The first one I know is the Peano existence theorem. I ...
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The category of discontinuous Banach spaces

A banach space is discontinuous if it is isometric to $DC(X)$ for some Hausdorff topological space $X$. ($DC(X)$ is defined here. We denote by $DBan$, the category of all discontinuous ...
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Norm condition in a Banach lattice

Consider the following "condition (J)" on the norm of a (real or complex) Banach lattice $E$: whenever $x$ and $y$ are disjoint (i.e., $|x|\wedge |y|=0$) then $\|x+y\|+\|x-y\|=2\|x\|+2\|y\|$. ...
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$C^*$-algebra generated by those operators that are bounded on every $\ell_p$

Suppose $T: c_{00} \to c_{00}$ is a linear map such that, when regarded as an infinite matrix, there is a uniform bound on the $\ell_1$-norms of its columns, and a uniform bound on the $\ell_1$-norms ...
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Banach space of discontinuous functions on a product space

Edit: According to comments of Eric Wofsy and Yemon Choi I edit the question. For a (compact) topological space $X$, we put $A=\{f:X\to \mathbb{C}\mid f\text{ is bounded}\}$. We define a ...
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Dual or pre-dual of BV

Was there any relevant work to determine the dual (or more likely the predual) of the space of bounded variation functions $BV(\mathbb{R}^n)$ (I recall the definition : a function in ...
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Book and Papers for properties of uniformly convex and locally uniformly convex and strictly convex Banach spaces.

I am looking for reference books and research articles which cover analysis of uniformly convex and locally uniformly convex and strictly convex Banach spaces.
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Is there a simple direct proof of the Open Mapping Theorem from the Uniform Boundedness Theorem?

The Open Mapping Theorem, the Bounded Inverse Theorem, and the Closed Graph Theorem are equivalent theorems in that any can be easily obtained from any other. The Closed Graph Theorem also easily ...
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286 views

Banach space of discontinuous functions(Killing continuous functions)

Edit: According to the comment of Prof. Majer, I revise the question: For a metric space $X$, we put $A=\{f:X\to \mathbb{C}\mid \text{f is bounded}\}$. We define two semi norm on $A$ ...
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556 views

When $C(X)$ is an injective $C(X)$-module? Current answer is erroneous

It is an old question if every injective Banach space is isomorphic as Banach space to $C(X)$-space. I would like to know if the weakened module version of this question is answered. More precisely: ...
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Can one show that the dual of a quasi-Banach space separates points without explicitly identifying the dual?

I'm interested in a question regarding the identification of some duals of quasi-Banach spaces. However, I'm not familiar with the quasi-Banach literature, so I'm hoping somebody can point me in the ...
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Banach space modulo a one-dimensional subspace =?

My question is the following: Given an infinite dimensional Banach space $E$ and a one-dimensional linear subspace $F\subset E$. It is well-known that this one-dimensional linear subspace is closed ...
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Reflexive subspaces of non-separable abstract $L_1$ spaces

An abstract $L_1$ space is a Banach lattice $E$ such that $\|x+y\|=\|x\|+\|y\|$ for disjoint $x,y\in E$. The space $L_1[0,1]$ is a separable example that contains subspaces isomorphic to $L_p[0,1]$ ...
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Large almost equilateral sets in finite-dimensional Banach spaces

Question: Does there exist a function $C:~(0,1)\to (0,\infty)$ such that for each $\varepsilon\in(0,1)$ every Banach space $X$ of dimension $\ge C(\varepsilon)\log n$ contains an $n$-point set ...
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Continuity in Banach space for non-linear maps

I want to find an example of a Banach space $X$ and a continuous map $f:X\rightarrow X$ such that $f$ is not bounded on the unit ball. I do not doubt that such an example exists, but I cannot make it ...
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164 views

Necessary conditions for optimality in Banach spaces

Let $X$ denote the non-negative "orthant" of the Banach space $L^2$ (or whatever you call the set of functions in $L^2$ that are non-negative), and let $C$ be a closed, convex subset of $X$. Let $f$ ...
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Counterintuitive consequences of the Hahn-Banach theorem

The axiom of choice has many counterintuitive consequences like the Banach-Tarski paradox. The Hahn-Banach theorem is a consequence of the axiom of choice, but it is weaker. I would like to know ...
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174 views

Aronszajn measure

In a geometric measure theory (GMT) course I'm following this year, the professor told us about the Aronszajn measure, and asked us to go check by ourselves what it reprensents (the course was about ...
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178 views

James $\ell_1$-theorem

This question was asked at MSe before but with no answer. I am struggling with the very last estimate in the proof of James' $\ell_1$-theorem. (Please see below an excerpt from Albiac and Kalton's ...