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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10
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1answer
398 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
219 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
160 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 ...
1
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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 ...
1
vote
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
votes
1answer
953 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
votes
1answer
3k 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 ...
62
votes
4answers
3k views

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
votes
1answer
197 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 ...
8
votes
1answer
208 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 ...
3
votes
2answers
300 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 ...
15
votes
4answers
479 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
257 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 ...
29
votes
2answers
8k views

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
221 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 ...
1
vote
1answer
291 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?
7
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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
votes
1answer
314 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
177 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 ...
4
votes
3answers
297 views

reflexive Banach space

Let $X$ be a Banach space and $B_X$ be the unit ball. Suppose that for each $\lbrace C_n\rbrace_{n=1}^\infty\subset B_X$ satisfying $C_n$ are closed convex and $C_n\supset C_{n+1}$ has a nonempty ...
-1
votes
1answer
183 views

Operators from $c_{0}$

My question seems to be easy but I cannot spot the answer. I am interested in ranges of operators defined on $c_0$. The celebrated "operator version" of Sobczyk's theorem says that if we are given a ...
0
votes
1answer
47 views

Range of operator being a limit

Let $X$ be a Banach space and let $T_n\colon X\to X$ be a family of bounded operators convergent to some operator $T\colon X\to X$. Is it true that $T(X)\subseteq \sum_{n=1}^\infty T_n(X)$? I mean ...
11
votes
1answer
693 views

Does separability follow from weak-* sequential separability of dual space?

Let $E$ be a Banach space. Suppose that $E'$ is weakly-* sequentially separable, that is, that there exists a countable $D \subset E'$ s.t. every $x' \in E'$ is a limit point of a ...
8
votes
1answer
749 views

Do there exist closed subspaces $X$, $Y$ of Banach space, such that $X+Y$ is not closed?

I am looking for an example of two closed subspaces of a Banach space, such that their sum is not closed.
3
votes
2answers
619 views

projection operators on topological vector spaces

Suppose $A \in \mathbb{R}^{m\times n}$. Then there exists a projection matrix $P$ onto the range of $A$. In other words, there exists a matrix $P \in \mathbb{R}^{m\times m}$ such that $P^2=P$, and ...
13
votes
1answer
2k views

Strong and weak convergence in $\ell^1$

Let $\ell^1$ be the space of absolutely summable real or complex sequences. Let us say that a sequence $(x_1, x_2, \ldots)$ of vectors in $\ell^1$ converges weakly to $x \in \ell^1$ if for every ...
6
votes
2answers
1k views

On the weak closure

Let $\lbrace e_n \rbrace$ for the standard unit vectors in $l_2$. I want to show that $0$ is in weak closure of $\lbrace\sqrt{n}e_n\rbrace$ but no subsequence of $\lbrace \sqrt{n}e_n\rbrace$ weakly ...
5
votes
2answers
1k views

Convergence of Cesàro means

Let $\lbrace x_n\rbrace\in l_2$ be a sequence weakly converges to $x$. I want to prove that there is a subsequence $\lbrace x_{n_k}\rbrace $ such that the Cesàro means ...
3
votes
1answer
254 views

weak$^*$-separability of $l_\infty^*$.

Where can I find the proof that $l_\infty^*$ is weeak$^*$-separable? I want to re-examine the proof of that fact.
4
votes
1answer
326 views

A closed subspace of $c_0$

Does anyone know an example of an infinite dimensional closed linear subspace $S$ of $X=c_0$ (with the sup norm) which is not isomorphic to $X$, i.e. there does not exist a linear one-to-one map $T$ ...
5
votes
2answers
425 views

Weak limit of an $L^1$ sequence

We have functions $f_n\in L^1$ such that $\int f_ng$ has a limit for every $g\in L^\infty$. Does there exist a function $f\in L^1$ such that the limit equals $\int fg$? I think this is not true in ...
32
votes
2answers
4k views

Connections between metrics, norms and scalar products (for understanding e.g. Banach and Hilbert spaces)

I am trying to understand the differences between $$ \begin{array}{|l|l|l|} \textbf{vector space} & \textbf{general} & \textbf{+ completeness}\\\hline \text{metric}& \text{metric ...
3
votes
1answer
112 views

Urysohn-like theorem in Banach spaces

I have a (separable) Banach space $E$ and two closed disjoint sets $F$, $G$ in $E$. Now I wish to prove the existence of a $C^2$-function (Fréchet differentiable) $f:E \to \mathbf R$ that is $1$ on ...
0
votes
1answer
410 views

Invertible bounded linear operators and closure

Let $T$ be an element of $B(X,X)$ (the bounded linear operators from X to itself), and let $W$ be a subset of $X$. Show that $T(\overline{W}) \subset \overline{T(W)]}$. Furthermore, if $T$ has bounded ...
9
votes
1answer
1k views

Vector, Hilbert, Banach, Sobolev spaces

Trying to wrap my head around all these different spaces. Which one is the most general? Can you summarize the differences between them? Is there a notable space that I missed?
4
votes
1answer
537 views

Proof of the Goldstine theorem

I have a question concerning the proof of the Goldstine theorem, that the image of the unit ball $B_X$ of some Banach space $X$ is dense in the unit ball $B_{X^{**}}$ of its bidual under the canonical ...
10
votes
3answers
956 views

Applications of the Hahn-Banach Theorems

Question: What are some interesting or useful applications of the Hahn-Banach theorem(s)? Motivation: Most of the time, I dislike most of Analysis. During a final examination, a question sparked my ...
14
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1answer
881 views

Different versions of Riesz Theorems

In Wikipedia, there are three versions of Riesz theorems: 1 The Hilbert space representation theorem for the (continuous) dual space of a Hilbert space; 2 The representation theorem for ...
46
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1answer
2k views

Example of a compact set that isn't the spectrum of an operator

This question is a follow-up to this recent question and related to that one. Is there an easy example of an (infinite-dimensional) Banach space $X$ and a non-empty compact set $K \subset ...
13
votes
1answer
2k views

Operator norm and tensor norms

I have a linear operator $A\in\mathcal{L}(X,Y)$ where $X$ and $Y$ are some Banach spaces (or Hilbert spaces would also do, if that simplifies the answer.). The operator norm of $A$ is given by $$ ...
2
votes
1answer
237 views

A question about why a space under a certain norm is complete

A theorem I am reading (about the existence and uniqueness of solutions to Sturm-Liouville intial-value problems) defines a space $B$ consisting of the continuous functions defined on a closed real ...
22
votes
2answers
1k views

Norm for pointwise convergence

Does there exist a norm on the space of all real-valued functions on the real line (or on an open set? a compact set?) such that convergence in this norm is equivalent to pointwise convergence?
12
votes
1answer
607 views

Are the coordinate functions of a Hamel basis for an infinite dimensional Banach space discontinuous?

The question is in the title really, but I suppose I could at least fix some notation here. Let $X$ be an infinite dimensional Banach space - over the reals for the sake of concreteness. Use choice ...
1
vote
1answer
370 views

Closedness of the image of the closed unit ball under a linear operator from a reflexive Banach space to an arbitrary Banach space

Let $V$ be a reflexive banach space. If $W$ is a Banach space and if $T$ is in $L(V,W)$, show that $T(B)$ is closed in $W$ where $B$ is closed unit ball in $V$, the problem is in the chapter of weak ...
5
votes
1answer
175 views

Why $L^{r}(X)\cap L^{t}(X)\subset L^{s}(X)$ for $1<r<s<t$?

I am working on this homework problem, and I am totally stuck: Let $(X,\mu)$ be a measure space, and let $1 \leq r < s < t < \infty$. Prove that there exist constants $\alpha,\beta>0$ so ...