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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Subspaces of $l_{1}$ are not Lipschitz complemented in $l_{1}$

I have thought about the following question for several years. This question may be stupid or not interesting. My question is: Is there a subspace $U$ of $l{_1}$ such that the quotient $l_{1}/U$ is ...
13
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1answer
272 views

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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83 views

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 ...
2
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1answer
129 views

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. ...
4
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165 views

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 ...
3
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1answer
301 views

Differentiability: Partially Defined Functions

These ideas came to my mind while reading Lee's Introduction to Smooth Manifolds. (Cf. discussion on p. 45.) Definition Let $E$ and $F$ be two Banach spaces together with a plain subset ...
2
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3answers
475 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 ...
1
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1answer
102 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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82 views

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 ...
6
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1answer
435 views

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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67 views

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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68 views

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 ...
5
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1answer
260 views

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?
7
votes
1answer
537 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: ...
6
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4answers
507 views

Example of noncomplete quotient of complete lcs mod closed subspace

The following statement is well-known: for a Fréchet space $V$ and a closed subspace $W \subseteq V$ the quotient $V / W$ is again complete and hence a Fréchet space. For the particular case of a ...
2
votes
1answer
198 views

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 ...
5
votes
3answers
344 views

Structure of an intersection of $L^p$-spaces

In what follows, $L^p$ denotes the space of functions from $\mathbb{R}$ to $\mathbb{R}$ such that $\int_{\mathbb{R}} |f(x)|^p\mathrm{d}x < \infty$. I am interested to understand the structure we ...
7
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1answer
138 views

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 ...
0
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0answers
83 views

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 ...
0
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0answers
46 views

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] ...
4
votes
2answers
178 views

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 ...
6
votes
0answers
352 views

Second duals of Grothendieck spaces

The classical example of a Grothendieck space is $\ell_\infty$. It is also known that its even duals $\ell_\infty^{**}$, $\ell_\infty^{(4)}$, $\dots$ are Grothendieck spaces. (See, e.g., this note ...
7
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0answers
93 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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0answers
68 views

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 ...
2
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1answer
160 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 $f$ ...
3
votes
0answers
85 views

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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0answers
230 views

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 ...
0
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0answers
57 views

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 ...
7
votes
2answers
181 views

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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0answers
97 views

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 ...
3
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104 views

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 $\|x+y\|+\|x-y\|=2\|x\|+2\|y\|$. ...
13
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0answers
256 views

$C^*$-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 ...
9
votes
1answer
357 views

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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2answers
97 views

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.
2
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1answer
276 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 $A$ ...
10
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2answers
495 views

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 ...
17
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0answers
260 views

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 ...
23
votes
2answers
1k views

What theorem constructs an initial object for this category? (Formerly “Integrability by abstract nonsense”)

Tom Leinster has a note here about how you can realize L^1[0,1] as the initial object of a certain category. You should really read his note because it is only 2.5 pages and is much more charming than ...
5
votes
1answer
158 views

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 $L_p[0,1]$ ...
7
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0answers
349 views

Minkowski's Inequality for Integrals in Orlicz spaces

EDIT: I have changed the question to have less parameters, fitting it into the context of Orlicz spaces. Suppose $f:[0,\infty)\to[0,\infty)$ is convex and increasing, ...
4
votes
4answers
332 views

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 ...
5
votes
2answers
255 views

Contractively complemented subspaces of $c_0(I)$

Does every contractively complemented subspace of $c_0(I)$ is isometric to $c_0(J)$ for some $J\subseteq I$? May be someone has a counterexample?
9
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945 views

Boundedness of nonlinear continuous functionals

Let $K$ be the closed unit ball of $C[0,1]$, and let $f$ in $C(K,\mathbb{\, R})$. Is it true that there exists an infinite dimensional reflexive subspace $E$ of $C[0,1]$ s.t. $f(K\cap E)$ is bounded ? ...
0
votes
1answer
172 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 ...
9
votes
1answer
623 views

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 ...
1
vote
1answer
172 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 ...
5
votes
0answers
208 views

Approximating a measurable function from a second-countable, locally compact Hausdorff group to a separable Banach space

Let $ G $ be a second-countable, locally compact Hausdorff group and $ B $ a separable Banach space. We say that a function $ f: G \to B $ is Bochner-measurable if and only if it is the everywhere ...
3
votes
2answers
193 views

Is the space of trace class operators finitely representable in an $L^1$-space?

I am interested in knowing whether the space of trace class operators is (crudely) finitely representable in an $L^1$-space. I suspect that the answer is negative but I am unable to find any argument ...
3
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2answers
242 views

Generalized Hardy-Littlewood-Sobolev Inequality

The Hardy-Littlewood-Sobolev Inequality says that $$\text{for $p,q,r\in (1,+\infty)$ such that }\quad 1-\frac1p+1-\frac1q=1-\frac1r,\tag {$\sharp$} $$ $$ \exists C, \forall u\in L^p(\mathbb ...
2
votes
2answers
276 views

If $ F(x,\bullet) \in {L^{2}}(G,B) $ for all $ x \in G $, then is $ x \mapsto F(x,\bullet) $ strongly measurable?

This question is related to something that I asked yesterday: If $ F(x,\bullet) \in {L^{\infty}}(G,B) $ for all $ x \in G $, then is $ x \mapsto F(x,\bullet) $ strongly measurable? Pietro Majer ...