# Tagged Questions

**-4**

votes

**1**answer

87 views

### A criterion of norm null sequences in Banach space [on hold]

I would like to know if for a weak* null sequence $\left( f_{n}\right) $ in a Banach space $X$,
the following characterisation is true and what about its proof:
$\left( f_{n}\right) $ is norm null ...

**1**

vote

**0**answers

46 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 ...

**3**

votes

**1**answer

79 views

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

Let $ (X,\Sigma,\mu) $ be a $ \sigma $-finite measure space and $ B $ a Banach space. A function $ f: X \to B $ is said to be strongly $ \mu $-measurable iff it is the almost-everywhere pointwise ...

**6**

votes

**1**answer

155 views

### Extending compact operators

Let $X$ be a separable, infinite-dimensional complex Banach space and $Y\subseteq X$ an infinite-dimensional closed subspace. Suppose $K:Y\to X$ is an arbitrary compact operator. I would like to ...

**3**

votes

**1**answer

198 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 ...

**-2**

votes

**0**answers

36 views

### Bounding and continuity in Banach spaces [closed]

Prove that if mapping in Banach space is bounded, it is continuous.

**5**

votes

**0**answers

85 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

104 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 ...

**2**

votes

**0**answers

68 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

118 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

86 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 ...

**0**

votes

**0**answers

156 views

### What is between super-reflexivity and reflexivity?

would you please tell me if there are properties or examples which are more than reflexivity and less than super-reflexivity?

**3**

votes

**1**answer

152 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

87 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 ...

**2**

votes

**1**answer

180 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 ...

**0**

votes

**0**answers

76 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 ...

**1**

vote

**0**answers

66 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

**0**answers

88 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 ...

**3**

votes

**2**answers

220 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 ...

**7**

votes

**0**answers

296 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

167 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 ...

**0**

votes

**0**answers

46 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

118 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

82 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

168 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

153 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

101 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 ...

**4**

votes

**0**answers

167 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

106 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

147 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, ...

**3**

votes

**1**answer

200 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

**1**

vote

**2**answers

131 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 $$\lim_{nâ†’âˆž}\|x_nâˆ’y_n\|=0$$ whenever $$x_n,y_nâˆˆSX$$ are such that $$\lim_{nâ†’âˆž}\|x_n+y_n\|=2$$ and there is a $zâˆˆX$ ...

**2**

votes

**1**answer

167 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

179 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 ...

**4**

votes

**0**answers

421 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

107 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

245 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 ...

**3**

votes

**1**answer

220 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

315 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 ...

**10**

votes

**1**answer

327 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

146 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 $Id_Y - ST \in \mathbb{K}(Y)$ ...

**-4**

votes

**1**answer

154 views

### Is there always a subspace of $L^p$ isomorphic to direct sums of $\ell^2,\ell^p$? [closed]

It is known that each $L^p$ (on a space with finite measure like $[0,1]$) $p>1$ space contains an isomorphic complemented copy of $\ell^2$ and $\ell^p$. I think this is the Kadets-Pelczynski ...

**2**

votes

**0**answers

165 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 ...

**3**

votes

**2**answers

216 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

418 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

56 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 UMDâˆ’property. Let $N\in\mathbb N$ and $s\in\mathbb R$ and $1\le p\le +\infty$ . ...

**1**

vote

**1**answer

171 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 ...

**4**

votes

**0**answers

141 views

### Ultrapowers and complemented subspaces

Let $Y$ be a closed subspace of a Banach space $X$, and let $\mathcal{U}$ be a nontrivial ultrafilter on the set $\mathbb{N}$ of all integer numbers.
It is not difficult to see that if $Y$ is ...

**5**

votes

**1**answer

169 views

### Introducing a dual space structure

Suppose that we have a Banach space $X$ together with some locally convex Hausdorff topology on $X$, weaker than the one given by the norm, which makes the unit ball of $X$ compact. Is $X$ ...

**3**

votes

**1**answer

162 views

### A differentiable version of the Michael selection theorem

Assume that $X$ and $Y$ are Banach spaces and $T:X\to Y$ is a bounded surjective linear map.
Is there a Gateaux differentiable function $g:Y\to X$ such that $T\circ g=Id_{Y}$?