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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3
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Balls in the space of bounded operators on a Hilbert space

Suppose $\mathsf{H}$ is an infinite-dimensional (non-separable preferably) Hilbert space. Consider the space $L(\mathsf{H})$ of all bounded operators on it. Is there $0\neq W\in L(\mathsf{H})$ such ...
2
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0answers
134 views

Embedding $\ell^\infty(\Gamma)$ into $\mathcal{B}(E)$

Is there any criterion answering the question: Let $E$ be a Banach space. When does the Banach space $\mathcal{B}(E)$ of all bounded operators on $E$ contain a copy of $\ell^\infty(\Gamma)$? Here ...
11
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2answers
3k views

Compact operator maps weakly convergent sequences into strongly convergent sequences

I found the following property of compact operators in a proof, and I can't prove it. Prove that if $T \in \mathcal{L}(E,F)$ is compact, and if $u_n \rightharpoonup u$ (the sequence converges ...
3
votes
1answer
876 views

Young's inequality for discrete convolution

Young's inequality for convolution of functions states that for $f\in L^p(\mathbb{R}^d)$ and $g\in L^q(\mathbb{R}^d)$ we have $$\|f\star g\|_r\le\|f\|_p\|g\|_q$$ for $p$, $q$, $r$ satisfying ...
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2answers
176 views

Extended James space

The discussion in Convergence in topologies, especially the comments of GEdgar, led me to another (converse) question concerning convergence. In the paper G. A. Edgar, A long James space, in: ...
11
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1answer
453 views

Renorming $\mathcal{B}(\mathcal{H})$?

Consider the Banach space of all bounded operators $\mathcal{B}(\mathcal{H})$ on a (separable if you wish) Hilbert space $\mathcal{H}$ with the operator norm. Can we renorm this space to a strictly ...
2
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1answer
117 views

Is a Fréchet differentiable map between complex Banach spaces locally given by a “power series”?

Let $X,Y$ be Banach spaces over $\mathbb{C}$ and let $U \subset X$ be open. If $f:U \to Y$ is Fréchet differentiable at every point of $U$, can we locally expand $f$ as a "power series"? To be more ...
6
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1answer
3k views

Inequality between $\ell^p$-norms

Suppose that a sequence $x=(x_n)$ belongs both to $\ell^p$ and $\ell^q$ ($p,q>1$, $p\neq q$). Is there any inequality between $\|x\|_p$ and $\|x\|_q$. Can one $\ell^p$ be continuously embedded into ...
1
vote
1answer
93 views

Existence of extremally disconnected space $X$ for which $C(X)$ is $\ell_1^4$

Can one view $\ell_{1}^{4}$, $\mathbb{R}^{4}$ equipped with the $\ell_{1}$-norm, as a space of continuous functions on any extremally disconnected space?
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1answer
277 views

Extremally disconnected space

Can one view $\ell_{\infty}^{4}$, $\mathbb{R}^{4}$ equipped with the $\ell_{\infty}$-norm, as a space of continuous functions on some extremally disconnected space? and what would the extremally ...
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2answers
1k views

Original proof of Uniform Boundedness Principle (Banach Steinhaus) and related questions

Can someone please provide me with any of the things listed below : a list of different proofs of (some version of) the Uniform Boundedness Principle (also known as the Banach Steinhaus theorem), I ...
4
votes
1answer
101 views

A nonreflexive module isomorphic to its double dual

I know that the definition of reflexive module is that the $R$-module $M$ should be isomomorphic to its double dual $M^{**}$ via the canonical map $M\rightarrow M^{**}$. I'd like to know an ...
9
votes
1answer
576 views

Isometric to Dual implies Hilbertable?

Let $X$ be a Banach space and suppose that $X$ is isometric to its continuous dual space $X^*$. Must $X$ be hilbertable in the sense that there exists an inner product which induces the norm on $X$? ...
8
votes
1answer
896 views

Taylor's theorem in Banach spaces

Let $f$ be a real function of a single real variable. Suppose that $f$ is $n$ times differentiable at some $x$, for some integer $n\geq 1$. Making no further assumptions, we have $$ f(x+h) = f(x) + ...
3
votes
1answer
174 views

Basis properties of the polynomial system in the space of continuous functions

Consider $\{1, t, t^2, t^3, \ldots\}$ as a subset of $C\big( [0, 1]\big)$. Clearly this system is linearly independent. Also, it is a complete system (meaning that its closed linear hull is the whole ...
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2answers
152 views

Quotient of $\ell_1$ by space of finite sequences

Consider $\Phi$ to be the space of sequences that have finitely many non-zero terms. The space is not closed in $\ell_1$, therefore $\ell_1/\Phi$ with the quotient topology is not Hausdorff, and so it ...
8
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1answer
558 views

Is the space of uniformly left continuous functions on [0,1] complete?

We'll say that a function on $[0,1]$ is uniformly left continuous if for every $\epsilon > 0$ there exists $\delta > 0$ such that $x \in (y - \delta, y)$ implies $|f(x) - f(y)| < \epsilon$ ...
6
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0answers
152 views

Linear isomorphisms with dense graph

Is it true that for each infinite dimensional Banach space $X$ there exists a linear bijection $f: X \rightarrow X$ with a dense graph? A graph of $f$ it is the set $\Gamma(f):=\{(x, f(x)): x \in X ...
2
votes
1answer
129 views

$d(T_n,T)\to 0$ if and only if $T_n\to T$ pointwise on the closed unit ball

I admit this is an homework. However i'm quite unused to this kind of argument so i would like to receive a suggestion or a confirm about my guesses.. So.. Let $X$ be a separable Banach space with ...
5
votes
2answers
563 views

Equivalent definitions of unconditional convergence

I am across two definitions for unconditional convergence for which it is not immediately obvious to me that they are equivalent. Here are the definitions. Throughout, $\frak{X}$ will denote a Banach ...
6
votes
1answer
520 views

If a subspace of $X^*$ separates points, is it weak-* dense?

Let $X$ be a Banach space, $X^*$ its dual. Suppose $E$ is a linear subspace of $X^*$ which separates points (i.e. if $f(x)=0$ for all $f \in E$, then $x=0$). Must $E$ be weak-* dense in $X^*$? In ...
2
votes
1answer
188 views

Squeeze theorem for duals of Banach spaces

Suppose that $X\subseteq Y \subseteq Z$ are Banach spaces such that $X$ is complemented in $Y$ and the duals $X^*, Z^*$ are isomorphic. Must the dual $Y^*$ be isomorphic to $X^*$?
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124 views

Injective tensor products of WCG spaces

The notion of a WCG space is a common roof for separable Banach spaces and reflexive ones. Nevertheless, the class is stable under $\ell^p(\Gamma)$-sums for any set $\Gamma$ when $p>1$ and ...
2
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2answers
269 views

Injection from non-separable to separable subspaces

Let $\Gamma$ be an uncountable set (possibly of cardinality $\aleph_1$). Is there an injective bounded linear operator $T\colon c_0(\Gamma)\to X$, where a) $X$ is some separable Banach space b) ...
2
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2answers
278 views

When is $\mathcal{L}_\infty$ a vector space?

Suppose that $X$ is a subset of a Hausdorff topological space, and $Z\subseteq X$ a member of the family of Borel subsets on $X$. Let $B(Z)$ the set of bounded functions on $Z$, equipped with the ...
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votes
2answers
412 views

Positive functionals on $\ell^\infty$

A continuous linear functional $\varphi: \ell^\infty \to \mathbb{R}$ is said to be positive if $$ x \ge 0 \rightarrow \varphi(x) \ge 0 \quad \forall x \in \ell^\infty.$$ If $\varphi$ is in $\ell^1 ...
10
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1answer
399 views

Proof of Hölder inequality by differentiation

I need a reference where we can read a proof of the inequality $\|f\|_r\leq \|f\|_p^{1-\theta}\|f\|_q^\theta$ where $\frac{1}{r}=\frac{1-\theta}{p}+\frac{\theta}{q}$ for $L^p$-spaces of a measure ...
5
votes
1answer
223 views

Comparison between Rademacher average and random average

Let $X$ be a Banach space. We let $j=e^{2i\pi/3}$. Let $(\epsilon_i)$ be a sequence of independent Rademacher variables on a fixed probability space $\Omega$. Let $(\varphi_i)$ be a sequence of ...
5
votes
3answers
381 views

Is $\ell^1$ isomorphic to $L^1[0,1]$?

Can there be a continuous linear map, with a continuous inverse, from $l^{1}$ to $L^{1}(m)$ where $m$ is the Lebesgue measure on the unit interval $\left[0,1\right]?$ My thinking to this should be ...
4
votes
1answer
162 views

One more predual space of the space of measures?

I am considering the Banach space $A$ with $\sup$-norm, which is the uniform closure of functions on a segment that are continuous but a finite collection of jump points, where they have limits from ...
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0answers
99 views

Are nuclear Montel spaces projective?

The well known fact about $\ell^1$ says that the Schauder basis of $\ell^1(I)$ behaves more-less like a Hamel basis, namely if $X$ is any Banach space and $\mathcal{E}=(e_i)_{i\in I}$ is a basis for ...
2
votes
1answer
171 views

Deviations of a bounded linear operator

Let $X$ be a Banach space, $\mathcal A:X\to X$ is linear and bounded in the norm $$ \|\mathcal A\| = \sup\limits_{x\in X}\frac{\|\mathcal Ax\|}{\|x\|}. $$ Suppose that an equation $$ x = \mathcal ...
5
votes
1answer
253 views

Showing that $l^p(\mathbb{N})^* \cong l^q(\mathbb{N})$

I'm reading functional analysis in the summer, and have come to this exercise, asking to show that the two spaces $l^p(\mathbb{N})^*,l^q(\mathbb{N})$ are isomorphic, that is, by showing that every $l ...
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3answers
252 views

Is a contraction semigroup infinitesimal operator bounded?

Let $T_t:L\to L$ be a semigroup of linear operators $T_t$ acting on a Banach space $L$. Assume that $$ \|T_t\| := \sup\limits_{f\in L}\frac{\|T_tf\|_L}{\|f\|_L} \leq 1 $$ for all $t\geq 0$. The ...
11
votes
1answer
2k views

Weak-to-weak continuous operator which is not norm-continuous

Can one give a "relatively easy" example of a linear mapping $T\colon X\to X$ ($X$ a Banach space) which is a) weak-to-weak continuous b) weak*-to-weak* continuous ($X=Y^*$) but not norm-to-norm ...
10
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1answer
954 views

Nonnegative linear functionals over $l^\infty$

My purpose is a clarification of the role of the axiom of choice in constructing limits for bounded sequences. Namely, we want a linear functional of norm 1 defined on the space of all bounded complex ...
41
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1answer
4k views

Was Grothendieck familiar with Stone's work on Boolean algebras?

In short, my question is: Was Grothendieck familiar with Stone's work on Boolean algebras? Background: In an answer to Pierre-Yves Gaillard's question Did Zariski really define the Zariski ...
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4answers
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Is it possible for a function to be in $L^p$ for only one $p$?

I'm wondering if it's possible for a function to be an $L^p$ space for only one value of $p \in [1,\infty)$ (on either a bounded domain or an unbounded domain). One can use interpolation to show that ...
6
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1answer
200 views

Projecting onto the diagonal given Banach spaces with unconditional bases

Let E be a Banach space with a 1-unconditional basis $(e_n)$ (for example, $\ell^p$). Then an operator T on E can be thought of as an infinite matrix, in the obvious way. Clearly each scalar on the ...
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1answer
209 views

Abelian sub-C*-algebras

Given a non-abelian C*-algebra $A$. I am wondering what are the possible abelian sub-C*-algebras of $A$. Let $K$ be the spectrum of $A$. Does $A$ contain an isomorphic copy (as a Banach space) of the ...
3
votes
2answers
131 views

Geometric Sums in Banach Algebra

Let $E$ be a Banach Algebra with identity, and $v\in E$, so that $||v|| < 1$. The geometric series $w = \sum_{k=0}^\infty v^k$ converges in the norm. I can show that $||w|| \le ...
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2answers
303 views

Is a complex space more “advanced” than a “generic” real space?

For instance, does taking the square root of a complex number and its complex conjugate create a metric that "automatically" makes it an inner product space? Is a complex space more complete than a ...
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484 views

Banach spaces over fields other than $\mathbb{C}$?

Sorry, this is a rather vague question. I was just wondering if there is any kind of theory about normed (if possible Banach) spaces over fields other than the real or complex numbers. I'm guessing ...
2
votes
0answers
258 views

How to prove Campanato space is a Banach space

Let $p\ge1$, $\mu\ge0$ and $\Omega$ be a bounded open set in $\mathbb{R}^n$. Campanato space embraces all $u$'s which ...
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votes
2answers
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The Duals of $l^\infty$ and $L^{\infty}$

Can we identify the dual space of $l^\infty$ with another "natural space"?. If the answer yes what about $L^\infty$. By the dual space I mean the space of all continuous linear functionals.
4
votes
1answer
222 views

Tensor products and vector valued functions

Given a non-empty set $S$ and a Banach space $X$. Let $B(S,X)$ be the space of all bounded maps from $S$ to $X$. Can we identify $B(S,X)$ with $\ell^\infty(S) \otimes X$, where $\otimes$ is some kind ...
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1answer
293 views

Complementability of von Neumann algebras

Is every von Neumann algebra complemented in its bidual? It is certainly true for commutative von Neumann algebras as their spectrum is hyperstonian. Is it 1-complemented?
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votes
3answers
2k views

Operator norm on product space

I have a bilinear operator $B\colon X \times Y\to Z$ with $X,Y,Z$ normed spaces, and define a norm on $X \times Y$ by $\lVert(x,y)\rVert = \lVert x\rVert_X + \lVert y\rVert_Y$ (using the respective ...
4
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1answer
317 views

Criterion for a bounded linear operator to be compact

Let $X,Y$ be Banach spaces. $T\colon X\to Y$ be a bounded linear operator. How can I prove that $T$ is compact if and only if there is $\lbrace x_n^*\rbrace\subset X^*$ such that $\|x_n^*\|\to 0$ and ...
6
votes
2answers
178 views

Covering theorem for Banach spaces

Are there any Banach space $X$ with $\operatorname{dim}(X)=\infty$ satisfying $S_X=\lbrace x\in X| |x|=1\rbrace $ is covered by for some $B_1,B_2,\ldots,B_N$, where $B_N$ are balls in $X$ with ...