**9**

votes

**1**answer

814 views

### Examples of continuous differential equations with no solution

Several theorems regarding differential equations are true not only in finite dimension but also in infinite dimension Banach spaces. This is in particular the case for the Cauchy–Lipschitz theorem.
...

**3**

votes

**0**answers

135 views

### A question on $p$-approximation property

We say that a subset $K$ of a Banach space $X$ is relatively $p$-compact ($1\leq p<\infty$) if there exists a $p$-summable sequence $(x_n)_{n=1}^{\infty}$ in $X$ such that
$$ K\subseteq ...

**2**

votes

**0**answers

174 views

### Finitely generated groups non-embeddable into $L_1(0,1)$

I am interested in finitely generated groups which, endowed with their word metrics, do not admit bilipschitz embeddings into $L_1(0,1)$. I know two classes of such groups:
(1) Heisenberg group ...

**11**

votes

**1**answer

253 views

### Embeddings of finitely generated groups into uniformly convex Banach spaces

de Cornulier, Tessera, and Valette (Geom. Funct. Anal. 17 (2007), 770-792) conjectured that a finitely generated group $G$ with its word metric admits a bilipschitz embedding into a Hilbert space if ...

**1**

vote

**0**answers

83 views

### Transferring locally uniformly convex norm by bounded linear operator from one Banach space to another

I'm trying to find a simple proof this theorem
Let $Y$ be a locally uniformly convex Banach space and $X$ be a Banach space and let $T$ be a bunded linear operator from $X$ into $Y$ such that for ...

**0**

votes

**1**answer

126 views

### Two questions about convex subsets of Hilbert Space

Let H be an infinite dimensional and separable Hilbert Space. Do there exist disjoint, closed and bounded subsets A,B of H which satisfy the following conditions? (1) Each of A,B is convex and has a ...

**4**

votes

**2**answers

249 views

### Is the ideal of functions vanishing at a set complementable in $C(X)$?

Let $X$ be a compact Hausdorff topological set, and $Y$ be its closed subset. Is the ideal of functions vanishing on $Y$
$$
I=\{f\in C(X):\ \forall y\in Y\ f(y)=0\}
$$
complementable (as a closed ...

**3**

votes

**0**answers

195 views

### Decreasing sequence of closed convex sets in a Banach space

Let $(C_n)$ be a decreasing sequence of closed convex subsets in a Banach space $(E,\Vert \cdot \Vert)$. The question I have is about the content of $C=\bigcap_{n=0}^\infty C_n$.
If the $C_n$ are ...

**7**

votes

**0**answers

355 views

### The Banach space of bounded functions with countable support

Let $X$ be a set of cardinality $\aleph_1$ and consider the Banach space $\ell_\infty^c(X)$ of all scalar-valued bounded functions on $X$ which are non-zero only for countably many elements of $X$ ...

**3**

votes

**1**answer

253 views

### Positive definite quadratic forms on Banach spaces

This is a question about characterizing Hilbert spaces in terms of quadratic forms. Let $X$ be a real Banach space and $E$ a bounded quadratic form on it, it is called positive definite if ...

**5**

votes

**0**answers

198 views

### Large subspaces with small basic constants in finite-dimensional Banach spaces

Let $B\in(1,\infty)$. I am interested in estimates for the
function $f_B(n)$ defined as the largest $k\in\mathbb{N}$
satisfying the condition: Each $n$-dimensional Banach space
contains an ...

**2**

votes

**0**answers

59 views

### König-Wittstock norm on Banach space

Let $(E,\Vert.\Vert)$ be a real Banach space and $\ell\ne 0$
an non-continuous linear form on $E$.
Let $a\in E$ be such that $\ell(a)=1$.
König-Wittstock [Non-equivalent complete norms
and would-be ...

**1**

vote

**1**answer

94 views

### A formally weaker form of the extendable local reflexivity for Banach spaces

Rosenthal introduce the notion of the extendable local reflexivity for Banach spaces as follows: Let $X$ be a Banach space and let $\lambda\geq 1$. $X$ is said to be $\lambda$-extendably locally ...

**1**

vote

**2**answers

131 views

### The reflexivity of the space generated by a convex, balanced and compact set

Let $K$ be a convex,balanced and compact subset of a Banach space $X$. We let $X_{K}:=span\{K\}$. Define the norm $\|x\|_{K}:=\inf\{\alpha>0: x\in \alpha K\}, x\in X_{K}$. Then $(X_{K},\|\|_{K})$ ...

**10**

votes

**1**answer

304 views

### 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 ...

**1**

vote

**1**answer

223 views

### On injective Banach space $C_0(X)$

Let $X$ be a locally compact Hausdorff space, such that $C_0(X)$ is an injective Banach space, i.e. a $\mathfrak{P}_\lambda$ space for some $\lambda\geq 1$.
Is it true that $X$ is compact?
If ...

**2**

votes

**0**answers

171 views

### Clarkson's inequalities for Banach space valued functions

In standard analysis, Clarkson's inequalities expresses the norms of the sum and difference of two functions in $L^p$ in terms of the norms of the individual functions. In particular, one may use the ...

**2**

votes

**1**answer

92 views

### Existence of normal structure in strictly convex Banach spaces

Does there exists a strictly convex Banach space which is not uniformly convex and has normal structure ?

**2**

votes

**0**answers

184 views

### Measurability of a map that takes a functional to its composition with a linear operator

Let $(X,\Sigma_X)$ be a measurable space such that $\Sigma_X$ is countably generated.
Let $B_b(X)$ be the Banach space of all bounded $\Sigma_X$-measurable functions $X\to\mathbb{R}$ equipped with ...

**4**

votes

**0**answers

172 views

### a question about Tsirelson's space

NOTE: I asked this question over at math.stackexchange.com but got no answer or comments after 3 days, probably because it's a bit specialized. Hopefully it is interesting enough to ask over here.
...

**1**

vote

**0**answers

111 views

### A question about Smulian lemma

Smulian lemma says Let $(X, ||.||)$ be a Banach space and$(X^*, ||.||^*)$ and let $x\in S_X=\{x\in X:||x||=1\}$ then
(i) $||.||$ is Frechet diffrentiable at $x$ iff ...

**3**

votes

**1**answer

99 views

### Productivity of Corson's property (C)

H.H. Corson in [C] introduced the following version of Lindelöf property for convex closed subsets of Banach spaces:
A Banach space $X$ has property (C) if every family of convex closed subsets of ...

**4**

votes

**0**answers

295 views

### Denseness of finite rank operators in $\mathcal{B}(X,Y)$

Let $X$ and $Y$ be Banach spaces and let $\mathcal{B}(X,Y)$ be the space of bounded linear operators from $X$ to $Y$. As noted in the answers to a question on
...

**5**

votes

**0**answers

115 views

### Banach spaces admitting no proper quasi-affinity

I am interested in examples of Banach spaces $X$ satisfying the following two conditions:
(1) Every (continuous linear) injective operator $T:X\to X$ with dense range is surjective.
(2) $X$ is ...

**3**

votes

**1**answer

332 views

### Predual of a Direct Sum of Banach Spaces

This may be basic, and if it is I apologize, but I have found no references to it in literature. I would appreciate a reference at least if I am wrong. I have supplied background for those interested, ...

**4**

votes

**1**answer

240 views

### Frechet differentiable implies reflexive?

Let $X$ be a Banach space. Is it true that if dual norm of $X^*$ is Frechet differentiable then $X$ is reflexive?
Can any one help me?
thanks

**5**

votes

**1**answer

134 views

### Tokarev's theorem on Banach lattices which are Grothendieck spaces

When browsing the literature, I have found the following theorem of E. Tokarev:
Let $X$ be a Banach lattice with weakly sequentially complete dual space. Then for any Banach space $Y$, every ...

**4**

votes

**0**answers

81 views

### Non-reflexive Orlicz spaces

I am looking for a good reference on Orlicz spaces; I would appreciate any books treating this topic from the Banach-space perspective. For example, I would like to find a reference to the following ...

**1**

vote

**2**answers

160 views

### relation between of uniformly rotund in every direction and uniformly rotund and locally uniformly rotund

The norm of a Banach space $X$ is said to be uniformly rotund in every direction if $$\lim_{n→∞}\|x_n−y_n\|=0$$ whenever $$x_n,y_n∈SX$$ are such that $$\lim_{n→∞}\|x_n+y_n\|=2$$ and there is a $z∈X$ ...

**4**

votes

**2**answers

482 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**

votes

**1**answer

198 views

### Is $H^\infty$ a second dual space?

Let $H^\infty$ denote the Banach space of all bounded analytic functions on the open disc $\mathbb{D}$. It is easy to see that $H^\infty$ is a dual space. However, is there a Banach sapce $Y$ such ...

**5**

votes

**3**answers

259 views

### Sequential closure of a set: standard terminology, notation, and properties

Let $X$ be a topological vector space (or, perhaps, more generally uniform space). Let $A\subset X$ be a subset. Let $A^s$ denote the set of limits of all convergent sequences (I guess $A^s$ is called ...

**6**

votes

**2**answers

128 views

### How does one prove that $L_1(\mu)$ is weakly sequentially complete for any measure?

It is a theorem of Steinhaus that for any finite measure $\mu$, the Banach space $L_1(\mu)$ is weakly sequentially complete. Using the Radon-Nikodym theorem one can extend this easily to ...

**4**

votes

**0**answers

541 views

### On the projective tensor product of $c_{0}$ by $c_{0}$

Let $E$ be the projective tensor product of $c_{0}$ by $c_{0}$. Does it follow that $E$ is isomorphic to no subspace of $C(K)$, where $K$ is countable compact metric space?
When $C(K)$ is isomorphic ...

**1**

vote

**1**answer

123 views

### Interpolation and embeddings for parabolic function spaces

I have a somewhat easy looking question on parabolic function spaces:
Let $B$ be a ball in $\mathbb R^n$ and let $T>0$. Denote $Q:=B \times [0,T]$. Assume $f \in L^2(Q) \cap L^\infty(0,T; L^q(B))$ ...

**8**

votes

**0**answers

268 views

### Closure of $L(\ell^2,\ell^2)$ in $L(\ell^2,\ell^\infty)$

Let $\ell^2$, $\ell^\infty$ denote the usual sequence spaces and let $L(\ell^2,\ell^2)$ the Banach space of bounded linear operators from $\ell^2$ to $\ell^2$ as well as $L(\ell^2,\ell^\infty)$ the ...

**1**

vote

**0**answers

77 views

### Example of a Banach lattice satisfying some conditions

Let a Banach lattice $E$ satisfying in the following conditions:
The lattice operations of the dual $E'$ are weak* sequentially continuous.
$E$ is not $σ$-Dedekind complete.
Is there such Banach ...

**3**

votes

**1**answer

250 views

### Characterization of $l_p$ up to a linear isometry

There is a science called "Geometry of Banach spaces". I wonder if they managed to give a geometric characterization of $\ell_p$ ($p\in[1,\infty]$) up to isometric isomorphism (among all Banach ...

**6**

votes

**2**answers

334 views

### $\|T\|_2 \le \sqrt{\|T\|_1\|T\|_\infty}$

Let $T$ be a linear operator acting on a finite-dimensional real or complex
vector space. As a direct consequence (or rather a particular case) of the
Riesz-Thorin theorem, we have
$$ \|T\|_2 \le ...

**35**

votes

**3**answers

2k views

### Absolute value inequality for complex numbers

I asked this question on stackexchange, but despite much effort on my part have been unsuccesful in finding a solution.
Does the inequality
$$2(|a|+|b|+|c|) \leq |a+b+c|+|a+b-c|+|a+c-b|+|b+c-a|$$
...

**1**

vote

**1**answer

109 views

### Operators from $\ell_\infty$

It is well-known that non-weakly compact operators from $\ell_\infty$ into any Banach space act as isomorphisms on some subspace of $\ell_\infty$ isomorphic to $\ell_\infty$. I have a question in this ...

**13**

votes

**1**answer

463 views

### Operator norms of circulant matrices

The definition and basic properties of circulant matrices can be found here: http://en.wikipedia.org/wiki/Circulant_matrix.
For complex numbers $a_1,\ldots,a_n$, I will use the notation
$$
...

**0**

votes

**0**answers

162 views

### Can I define Fredholm Index using $\dim \ker ST - \dim \ker TS$?

$X$, $Y$ are Banach spaces.
Let $S \in L(X, Y)$, $T \in L(Y, X)$, where $L(X, Y)$ denotes the Banach algebra of bounded linear operators from $X$ to $Y$. If we have that $Id_Y - ST \in \mathbb{K}(Y)$ ...

**2**

votes

**1**answer

192 views

### Nuclear vs Integral operators on Hilbert spaces

Consider the Hilbert space $L_2=L_2[0,1]$. Is it true that for each nuclear (trace-class) operator on $L_2$ there exists a function $K\in L_1(L_2)$ such that
$$Tf = \int\limits_0^1 K(s) f(s) ...

**2**

votes

**0**answers

199 views

### Banach space interpolation theory in terms of categories

I have recently learned a bit about higher category theory.
And there is a question I would like to ask. Can we enrich banach space interpolation theory by using higher category theory?
Is it ...

**6**

votes

**3**answers

488 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 ...

**6**

votes

**0**answers

476 views

### Products of spaces containing no copies of $\ell_2(\Gamma)$

Given an infinite set $\Gamma$, I would like to know if the class of Banach spaces containing no copies of $\ell_2(\Gamma)$ is stable under finite products.
When $\Gamma$ is countable the answer is ...

**1**

vote

**0**answers

70 views

### Is $\partial^\alpha$ a map $H^{s,p}(\mathbb R^N,F)\to H^{s-|\alpha|,p}(\mathbb R^N,F)$?

More precisely, the question is formulated as follows. Let $F$ be an arbitrary Banach space, especially not having the UMD−property. Let $N\in\mathbb N$ and $s\in\mathbb R$ and $1\le p\le +\infty$ . ...

**0**

votes

**0**answers

82 views

### What's the dual space of $c_{0}^{\mathcal {A}}(X)$?

Suppose that $X$ is a Banach space and $({\mathcal {A}},\alpha)$ is a Banach operator ideal. A sequence $(x_{n})_{n=1}^{\infty}$ in $X$ is said to be ${\mathcal {A}}-$convergent to zero if there exist ...

**1**

vote

**1**answer

200 views

### Final topology of surjective linear map on Banach space

Let $L:X\rightarrow Y$ be a surjective linear map from Banach space $(X,||\cdot||_X)$ to vector space $Y$ and denote with $\tau_L$ the final topology on $Y$ induced by $T$.
Is $\tau_L$ equivalent ...