**3**

votes

**1**answer

242 views

### Is $L(\ell_2,\ell_2)$ dense in $L(\ell_2,c_0)$?

Let $\ell_2:=\{ x:\mathbb{N} \to \mathbb{R}: \sum_{j=1}^\infty x_j^2<\infty\}$ and
$c_0:=\{ x:\mathbb{N} \to \mathbb{R}: \lim_{j\to\infty}x_j=0,\, \sup_{j\in\mathbb{N}}|x_j|<\infty\}$ denote the ...

**4**

votes

**2**answers

344 views

### Are Banach space norms (up to equivalence) unique?

Here is a naive question: is a "completing" norm of a vector space unique (up to equivalence) or can one find a vector space and two non-equivalent norms $\|.\|$ and $|||.|||$ that both induce a ...

**5**

votes

**0**answers

111 views

### A Banach space with all Hilbertian subspaces complemeneted

Assume that $X$ is a Banach space in which every Hilbertian subspace is complemented (let's say that all the projections are uniformly bounded). What can we say about $X$? It has to be K-convex. By ...

**0**

votes

**1**answer

146 views

### Relationship between LlogL and Hardy spaces

I think that for positive, one-dimensional, periodic functions, the following statement is true:
$$
f\in L log L(\mathbb{T})\Leftrightarrow f\in H^1(\mathbb{T}),
$$
where
$$
LlogL=\{f\in ...

**5**

votes

**0**answers

213 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 ...

**4**

votes

**1**answer

148 views

### Infinite dimensional subspaces of $L^1$

Suppose that $X$ is an infinite dimensional subspace of $L^{1}$. In some cases it is true that $X$ contains an isomorphic copy of an infinite dimensional Hilbert space. However, it is not the case ...

**5**

votes

**1**answer

119 views

### The Notion of Strong Measurability for Separable Banach Spaces

Let $ (X,\Sigma,\mu) $ be a measure space and $ B $ a Banach space. According to my understanding, a function $ f: X \to B $ is said to be strongly $ \mu $-measurable if and only if it is the ...

**3**

votes

**2**answers

260 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 ...

**0**

votes

**0**answers

100 views

### About the Intersection of the nested sequence of Chebyshev centers of weakly compact convex sets

Let $K_0$ be a weakly compact convex subset of a Banach space $X$. For each $n\in\mathbb{N}$, let $K_n$ be the set of Chebyshev centers of the set $K_{n-1}$. Suppose $K_0$ has a normal structure. Is ...

**3**

votes

**2**answers

191 views

### Operators on $\ell_\infty(\Gamma)$ and almost disjoint families of subsets

It is well known that given an operator $T:\ell_\infty\to\ell_\infty$ such that $Tx=0$ for each $x\in c_0$ there exists an infinite subset $M$ of the positive integers so that $Tx=0$ for each $x\in ...

**2**

votes

**1**answer

98 views

### Two basic questions on $p-$summable sequences

Let $X$ be a Banach space and $(x_{n})_{n=1}^{\infty}$ be a $p-$summable sequence in $X$. My basic questions are the following:
For any $\epsilon>0$, is there a sequence ...

**2**

votes

**1**answer

205 views

### Chebyshev centres of a bounded closed convex set in a strictly convex Banach space

Suppose $X$ is a strictly convex Banach space. Does there exist a bounded closed convex set $K$ in $X$ such that the set of all Chebyshev centers $C(K)$ of $K$ is a proper subset of $K$ with diameter ...

**9**

votes

**1**answer

565 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.
...

**0**

votes

**0**answers

83 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 $K$ is contained in ...

**2**

votes

**0**answers

148 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 ...

**10**

votes

**1**answer

228 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

75 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

122 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 ...

**0**

votes

**0**answers

93 views

### Relation between modulus of smoothness and reflexivity

Baillon proved that if $X$ is a Banach space with $\rho'_X(0)<\frac{1}{2}$, then $X$ has the fixed point property (by $\rho_X(t)$
we denote the modulus of smoothness). My questions are as ...

**4**

votes

**2**answers

239 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

138 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

335 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

194 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

179 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 ...

**0**

votes

**0**answers

54 views

### Representation of piecewise functions using variational inequalities

I need some help about how to represent piecewise smooth functions defined over N subintervals by using variational inequalities. From a paper I'm analyzing I get:
"If we can find a function ...

**2**

votes

**0**answers

52 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

92 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

127 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 $(X_{K},\|\|_{K})$ ...

**9**

votes

**1**answer

261 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

211 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

141 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

87 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

169 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

162 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

108 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

86 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

222 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

217 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

218 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

122 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

63 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

153 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 $z∈X$ ...

**3**

votes

**1**answer

317 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

178 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

220 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

121 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

514 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

114 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

258 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 ...