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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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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1answer
157 views

How to prove this equality in $\ell_1$ [on hold]

Let $w$ be any element in $\ell_1$, and $(w_n)$, $(z_n)$ two bounded sequence in $\ell_1$ that converge to 0 pointwise. Then we have ...
2
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32 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 ...
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1answer
51 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 ...
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162 views
+50

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 ...
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2answers
106 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 $(X_{K},\|\|_{K})$ ...
2
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0answers
108 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 ...
5
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1answer
144 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
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1answer
150 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 ...
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2answers
102 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 $z∈X$ ...
2
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1answer
62 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 ?
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136 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. ...
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0answers
87 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
1answer
72 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 ...
3
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1answer
164 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
4
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140 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
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97 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
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1answer
123 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, ...
5
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1answer
100 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
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56 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 ...
2
votes
1answer
152 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
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3answers
159 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
2answers
96 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
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0answers
266 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 ...
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1answer
102 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 $f \in L^2(Q) \cap L^\infty(0,T; L^q(B))$ ...
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231 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
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1answer
204 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 ...
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67 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 ...
6
votes
2answers
304 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 ...
7
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1answer
294 views

Existence of injective operators with dense range

Given two separable (infinite dimensional) Banach spaces $X$ and $Y$, it is not difficult to show that there exists an injective (bounded linear) operator $T:X\to Y$ with range dense in $Y$. See S. ...
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432 views

How hard (P, NP, NP-hard) is it to compute Schur norms of matrices (as multipliers)?

Given a matrix $A\in M_n(\mathbb{C})$, I will denote by $||A||_\infty$ the operator norm of $A$, as seen acting on the Hilbert space $\mathbb{C}^n$. This makes $M_n(\mathbb{C})$ into a Banach space ...
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3answers
1k 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 $$2(|a|+|b|+|c|) \leq |a+b+c|+|a+b-c|+|a+c-b|+|b+c-a|$$ ...
14
votes
1answer
619 views

$(1+\epsilon)$-injective Banach spaces, complex scalars

It is well known that a real Banach space which is $(1+\epsilon)$-injective for every $\epsilon >0$ is already 1-injective (Lindenstrauss Memoirs AMS, 1964, download here). Using common ...
10
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1answer
310 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 $$ ...
1
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1answer
95 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 ...
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1answer
442 views

Extending Continuous Sublinear maps on dense subsets of a Banach space

Suppose X' and Y are Banach spaces and X is a linear subspace dense in X'. Let T be a continuous map of X to Y satisfying: (1) ||T(x+y)|| is less than or equal to ||T(x)||+||T(y)||. Please prove ...
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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 $Id_Y - ST \in \mathbb{K}(Y)$ ...
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votes
1answer
148 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
1answer
129 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
0answers
154 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
2answers
191 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 ...
5
votes
0answers
368 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 ...
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48 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 $1\le p\le +\infty$ . ...
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137 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 ...
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1answer
168 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 ...
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71 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 ...
3
votes
1answer
300 views

A useful criterion in vector integration

I would like to know the proof of the following theorem: Consider two Banach spaces $X\hookrightarrow Y$ and $1<p,q\leq\infty$. Let $(f_n)_{n\geq 0}$ be a bounded sequence in $L^q(I,Y)$ and let ...
5
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1answer
160 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
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1answer
157 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 $T\circ g=Id_{Y}$?
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89 views

Mixed Tsirelson Norm

A couple of days ago I posted this question on Mathematics Stack Exchange. Surprisingly, so far, I haven't received any answers or comments about it (besides my own possible answer). Maybe I can get ...