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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277 views

how to show that $c_0$ is complete

I want to show that the metric space $(c_0,d_\infty)$ is complete, where $c_0$ is the collection of all sequences $x\colon \mathbb N\to\mathbb R$ which tend to $0$. I have already shown that the space ...
1
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1answer
853 views

Show that this linear operator is surjective

Let $E$ be a Banach space, and $T$ is a linear operator on $E$, furthermore,it's assumed that $$\sup_{||x||=1}|f(T(x))|<\infty,\forall f\in E^*;$$ $$\inf_{||x||=1}\sup_{||f||=1}|f(T(x)|>0;$$ and ...
6
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1answer
278 views

How to prove that the space of convergent sequences is complete?

Let $X=\mathrm{Conv}(\mathbf R)$, the collection of all convergent sequences in $\mathbf{R}$. Is the normed space $(X,\|\cdot\|_\infty)$ complete?
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2answers
2k views

Compact operators and completely continuous operators

A compact operator between Banach spaces is an operator that maps bounded sets into relatively compact sets, while a completely continuous operator maps all weakly convergent sequences into convergent ...
2
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2answers
201 views

On weak closure again.

I worked in this problem too: On the weak closure But after that I could not think of anything that can help me about the $l_p$ case. I mean, $\{ n^{1/p} e_n \}$ has $0$ as a weak accumulation ...
4
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1answer
254 views

Compactness of unit ball in $\ell_\infty$ with a different norm

Consider the normed spaces (over the field of real numbers) $X=(\ell_\infty,\|\cdot\|_\infty)$ and $Y=(\ell_\infty,\|\cdot\|)$ where $$\|x\|=\sup_{n\in\mathbf{N}}\frac{|x_n|}{2^n}.$$ How can I show ...
2
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0answers
146 views

Convergence of a function in a metric space to its metric

Given a metric space $(\mathbb{A},d)$ with a metric $d$ being the Euclidean metric, if $\lim_{t \rightarrow \infty}||A_{t+1}-A_t||\rightarrow 0$ is a convergent sequence where $A$ is a matrix with the ...
5
votes
1answer
133 views

Random variables with the same distributions

Let $K$ and $L$ be locally compact Hausdorff spaces. Also, let $P$ be a Radon probability measure on $K$ so that $(K,P)$ is a probability space. I want to know, whether two random variables ...
4
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1answer
124 views

Symplectic Chart

I was reading the article "Symplectic structures on Banach manifolds" by Alan Weinstein. In this article there is one theorem, which is as following: If $B$ is a zero neighborhood in Banach space. ...
2
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2answers
183 views

necessary conditions to be relatively compact

Let $X$ be a Banach space and $A\subset X$ be bounded. Suppose that, for any $\varepsilon>0$, $\exists\; F_\varepsilon$ a subspace of $X$ of finite dimension such that ...
6
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2answers
211 views

Universal separable Banach algebras

The well-known Banach–Mazur theorem says that $C([0, 1])$ is a universal separable Banach space, in the sense that if $X$ is any separable Banach space then there is a map $f : X \to C([0, 1])$ which ...
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2answers
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Is the fundamental theorem of calculus independent of ZF?

By the fundamental theorem of calculus I mean the following. Theorem: Let $B$ be a Banach space and $f : [a, b] \to B$ be a continuously differentiable function (this means that we can write $f(x + ...
3
votes
1answer
200 views

Completeness of the space of bounded bilinear maps

Let $X$, $Y$ and $Z$ be Banach spaces, and consider the space $Bil(X\times Y,Z)$ of all bilinear maps $B:X\times Y\to Z$ such that $x\mapsto B(x,y)$ and $y\mapsto B(x,y)$ are bounded maps from $X$ to ...
6
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1answer
452 views

Rainwater theorem, convergence of nets, initial topology

I've stumbled upon a result called Rainwater's theorem a few times, it seems to be a very useful result in connection with weak convergence in Banach spaces. Rainwater's theorem. Let $X$ be a ...
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1answer
181 views

Question about Cauchy sequence in $C(K)$

In my lecture, I need prove the existence of the anyone Cauchy sequence $(f_n)_{n\in \mathbb{N}}$ belonging $C(K)$ with the norm ${\Vert \cdot \Vert}_{\infty}=\sup_{x \in K}|f(x)|$ where $K$ is a ...
6
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2answers
2k views

Bounded linear operator maps norm-bounded, closed sets to closed sets. Implies closed range?

Question Suppose $T:X\rightarrow Y$ is a continuous, injective linear operator between Banach spaces. Suppose, in addition, that $T$ maps norm bounded closed sets in $X$ to closed sets in $Y$. Then ...
3
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3answers
365 views

Question about Sobolev embedding theorem

The Sobolev embedding theorem as stated in my notes says that if we have $k > l + d/2$ then we can continuously extend the inclusion $C^\infty(\mathbb T^d) \hookrightarrow C^l(\mathbb T^d)$ to ...
2
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1answer
121 views

Cancellation of addition on convex sets

I recently found a question about a property of the Minkowski sums. However the question was not properly answered (it used a projection argument which might not be true in a general Banach space). I ...
2
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1answer
333 views

Are compacta in a complete infinite dimensional normed space nowhere dense?

Let $X$ be an infinite dimensional Banach space. I want to show that any compact subset $\varnothing\neq A\subset X$ is nowhere dense. I've been able to prove the statement for ...
6
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2answers
763 views

How to prove that the $L^p$ spaces are infinite dimensional

It is well-known that (given a measure space $(S,\mathcal A,\mu)$ and $1\le p\le\infty$) the Banach space $L^p(S,\mathcal A,\mu)$ has infinite dimension. Is there an easy way to proof this statement ...
2
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2answers
230 views

Spectrum of an element in sub-algebra: $\sigma_A(b)\setminus \{0\}\subseteq \sigma_B(b) \setminus \{0\}$

Please help me to prove this:(or give me some references for this.) Thanks very much! Let $A$ be a (unital) algebra and $B\subset A$ a (unital) sub-algebra. Then for all $b\in B$: ...
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1answer
150 views

Ball contained in a convex cone

Let $X$ be a Banach space. Let $C\subset X$ be a closed convex cone with nonempty interior. We denote by $B_X$ and $S_X$ the unit ball and the unit sphere of $X$ respectively. Let $e\in C\cap S_X$, ...
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2answers
1k views

Confused about which Hölder spaces are Banach

If $\Omega$ is an open set in $\mathbb{R}^n$, is the Hölder space $C^{k, \alpha}(\Omega)$ Banach? Or is it only that $C^{k, \alpha}(\overline{\Omega})$ is Banach, like with ordinary continuous ...
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643 views

Isomorphic embedding of $L^{p}(\Omega)$ into $L^{p}(\Omega \times \Omega)$?

Let $(\Omega,\mu)$ be a finite measure space such that $\mu(\Omega)=1$. Suppose $1\leq p \leq \infty$. Let $\psi \colon L^p(\Omega) \to L^p(\Omega \times \Omega)$ be the map which maps $f$ onto the ...
4
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1answer
166 views

Map bounded if composition is bounded

Let $X,Y,Z$ Banach spaces and $A:X\rightarrow Y$ and $B:Y\rightarrow Z$ linear maps with $B$ bounded and injective and $BA$ bounded. Prove that $A$ is bounded as well. If I knew that $B(Y)$ is ...
0
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1answer
129 views

How to `bound' $L^\infty$ by the constant function $1$

Let $(S,\mathcal A, \mu)$ be a measure space and consider the Riesz space $L^\infty=L^\infty(S,\mathcal A, \mu)$ (under point-wise ordering). Let $1_X$ denote the indicator function on $S$ (which is ...
5
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2answers
217 views

Boundary of $L^1$ space

Is there any rigorous or heuristic notion of boundary of $L^1$ that is studied? I mean something loosely like the collection of functions or distributions defined by $$\left\{f\notin L^1: f_n\to ...
3
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2answers
306 views

Shifting a function is continuous

I'm slightly puzzled by the following: if $g(t)$ is a function in $L^q(X)$ then we can show that $g(t-x)$ is continuous function of $t$, i.e. for $\varepsilon > 0$ we can find $\delta$ such that ...
12
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1answer
2k views

Space of Complex Measures is Banach (proof?)

How can we prove that the space of Complex Measures is Complete? with the norm of Total Variation. I have stuck on the last part of the proof where I have to prove that the limit function of a Cauchy ...
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2answers
57 views

Can I get a bound like $[u]_\alpha \leq C\lVert u\rVert_{C^0(S)}$?

I badly need a bound like $$[u]_\alpha = \sup_{S}\frac{|u(x) - u(y)|}{|x-y|^\alpha} \leq C\lVert u \rVert_{C^0(S)} = C\sup_S |u|$$ where $S$ is compact and $C$ is a constant not depending on $u$. Can ...
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4answers
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Non-closed subspace of a Banach space

Let $V$ be a Banach space. Can you give me an example of a subspace $W\subset V$ (sub-vectorspace) that is not closed? Can't find an example of that yet. Thanks!
1
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1answer
133 views

differentiability classes

I am beginning to study multidimensional calculus using Spivak's Calculus on Manifolds, and so far I understand the purpose of considering the classes $C^n$ to be twofold. First, being in $C^n$ is a ...
5
votes
1answer
258 views

Range of bounded operator is of first category

Let $T$ be a bounded operator from a Banach Space $X$ to a normed space $Y$ such that $T$ is not onto, but $R(T)\subset Y$ is dense. Prove that $R(T)$ is of first category and not no-where dense. ...
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0answers
160 views

False proof of existence of unique point in closed convex subset of uniformly convex space

Let $B$ be a uniformly convex Banach space. Let $K \subset B$ be a closed convex subset. Edit Thank you for your helpful comments. I am trying to show that if $K$ is a closed convex subset of $B$ ...
2
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0answers
426 views

Bound on inverse operator

Define $X = {C^{2, \alpha}}(U \times [0,T])$ and $Y = {C^{0, \alpha}}(U \times [0,T])$ where $U$ is some real interval. Let $F:X \to Y$ be a map. Let $DF(g):X \to Y$ be a bounded linear operator for ...
2
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1answer
283 views

Prove that this is a Banach space

Let $I=[0,1]$ and let $\displaystyle X:=\left\{f: I\times \mathbb R\to \mathbb R\colon \sup_{(t,x)}\frac{|f(t,x)|}{1+|x|}<\infty\right\}$. Prove that $X$, equipped with the norm $\displaystyle ...
4
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3answers
2k views

Is product of Banach spaces a Banach space?

If $X$ is a Banach space, then I want to know if $X\times X$ is also Banach. What is the norm of that space? So for example, we know $C^k(\Omega)$ is Banach and I have a vector $v = (u_1, u_2)$ where ...
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1answer
99 views

Proving convexity of this set in $\ell^2$

This is a follow-up to the question I posted earlier this week. Consider, for a fixed sequence $(a_n)_n\in\ell^2$ the subspace $$C=\{(x_n)_{n}\in\ell^2 : |x_n|\le a_n\text{ for all ...
10
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1answer
373 views

Every multiplicative linear functional on $\ell^{\infty}$ is the limit along an ultrafilter.

It is well-known that for any ultrafilter $\mathscr{u}$ in $\mathbb{N}$, the map\begin{equation}a\mapsto \lim_{\mathscr{u}}a\end{equation} is a multiplicative linear functional, where ...
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5answers
1k views

How to show that this set is compact in $\ell^2$

Let $(a_n)_{n}\in\ell^2:=\ell^2(\mathbb{R})$ be a fixed sequence. Consider the subspace $$C=\{(x_n)_{n}\in\ell^2 : |x_n|\le a_n\text{ for all }n\in\mathbb{N}\}.$$ According to the book [Dunford and ...
4
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1answer
317 views

Cancellation law for Minkowski sums

Let $(X,\|\cdot\|)$ be a Banach space and $A,B,C\subset X$ closed bounded non-empty convex subsets. Let $+$ denote the Minkowski symbol for addition. Does the $+$ satisfy: $$A+C\subset B+C\implies ...
2
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1answer
302 views

Convex function on Banach space

Let $(Y,\|\cdot\|)$ a Banach space and $b\colon Y\to \mathbb{R}$ a nonnegative convex function such that, for some $\mathcal{E}>0$, the set $\{y\in Y\,:\, b(y)<\mathcal{E}\}$ is nonempty and ...
4
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0answers
116 views

A question regarding vector spaces with partial order

$\newcommand{\N}{\mathbf{N}}$$\newcommand{\R}{\mathbf{R}}$ Let $X=(X,\leq)$ be a Riesz space (a lattice which is also an ordered vector space over $\R$). Define $X^+=\{x\in X\colon x\ge 0\}$. Are the ...
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0answers
292 views

Uniformly bounded operators and inverse function theorem questions

Suppose $F:X \to Y$ is a map from Banach spaces $X=\widetilde{C}^{k+2, \alpha}(S)$ to $Y = \widetilde{C}^{k, \alpha}(S)$ where $S = I \times [0,T].$ Suppose the derivative $DF(u):X \to Y$ exists and ...
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vote
1answer
45 views

Gaussian type and Euclidean sections

I have a second question about Chapter 9 in Milman and Schechtman's book "Asymptotic theory of finite dimensional normed spaces" (first question here). It's about the proof of Theorem 9.7 (pg. 55). ...
5
votes
2answers
430 views

Uniform boundedness principle statement

Consider the uniform boundedness principle: UBP. Let $E$ and $F$ be two Banach spaces and let $(T_i)_{i \in I}$ be a family (not necessarily countable) of continuous linear operators from $E$ into ...
2
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1answer
162 views

A simple question about the open mapping theorem

$X, Y $ : Banach space, $T : X \to Y$ : linear bounded operator, onto. I'm studying open mapping theorem, but how can I prove this? If $B_Y (0, \epsilon_1 ) \subset \overline{T(B_X (0, \epsilon_2 ...
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1answer
191 views

Parabolic PDE $\to$ ODE on Banach space

Would someone please explain to me the concept of converting a parabolic PDE to an ODE on Banach space? If I have a PDE, say $$u_t = f(u_{xx}, u_x, u, p)$$ where $p$ is a parameter and the solution ...
2
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1answer
330 views

A question regarding convergence of sequences of $L_p$ functions

Let $(X,\mathcal M, \mu)$ be an arbitrary measure space and $1\le p<\infty$. I am curious whether the following statement holds: Let $\{f_n:X\to\mathbb{R}:n\in\mathbb{N}\}_n$ be a sequence in ...
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1answer
121 views

Where to find information on the Hilbert cube in $\ell^2$

The Hilbert cube $H$ in $\ell^2=\ell^2(\mathbb{R})$ is the subset given by $$H=\lbrace(x_n)=(x_1,x_2,\ldots)\in\ell^2:|x_n|\le2^{-n} \text{ for }n=1,2,\ldots\rbrace.$$ I've heard that ...