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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2
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411 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
250 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 ...
3
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3answers
1k 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 ...
1
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1answer
94 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
332 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 ...
8
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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
votes
1answer
301 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
votes
1answer
282 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
votes
0answers
110 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 ...
1
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0answers
276 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 ...
1
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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
398 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
votes
1answer
149 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 ...
1
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1answer
183 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
votes
1answer
275 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 ...
1
vote
1answer
105 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 ...
1
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1answer
170 views

norm of operator in Hilbert space and complex conjugate Banach space

Let $E$ and $F$ be complex Banach spaces. We denote by $\overline{E}$ the compex conjugate of $E$, that is, the vector space $E$ with the same norm but with the conjugate multiplication by a complex ...
3
votes
3answers
188 views

Question about proof that multiplication in Banach algebra is continuous

Here's the proof in my notes: Where does the last inequality come from? If I want to show that it's continuous at $((x,y)$ I can use the inverse triangle inequality to get $$ (\|x^\prime\| + ...
1
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2answers
282 views

Composition of continuous and closed operators is closed

Let $X,Y,Z$ Banach spaces, $\text{dom}(S)\subset Y$, let $T:X\rightarrow Y$ be linear and continuous and let $S:\text{dom}(S)\rightarrow Z$ be linear and closed. Show that the composition $ST$ is ...
1
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3answers
132 views

$B(V,W)$ is complete if $W$ is

Let $B(V,W)$ be the space of bounded linear maps from $V$ to $W$. Then it is complete with respect to the operator norm. Can you tell me if my proof is correct? Thanks. It's easy to verify that the ...
6
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2answers
145 views

Inclusion of $\mathbb{L}^p$ spaces, reloaded

I have a follow-up from this question. It was proved that, if $X$ is a linear subspace of $\mathbb{L}^1 (\mathbb{R})$ such that: $X$ is closed in $\mathbb{L}^1 (\mathbb{R})$; $X \subset \bigcup_{p ...
5
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2answers
194 views

Are countable intersections of convex sets convex?

Let $X$ be a Banach space $\{C_n\colon n\in\mathbb N\}$ a collection ofconvex sets in $X$. Is the set $$C=\bigcap_{n\in\mathbb N}C_n$$ convex?
1
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1answer
76 views

Euclidean sections of normed spaces with known cotype

I'm having trouble digesting the proof of Theorem 9.6 in Milman and Schechtman's classic book "Asymptotic theory of finite dimensional normed spaces" (pg. 54). I'm new to functional analysis, so this ...
0
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0answers
275 views

$C_c(X)$ dense in $L^p$

In class we proved that $C_c(X)$ is dense in $L^p$ where $X$ is a locally compact, $\sigma$-compact Hausdorff space either equipped with a Radon measure or equipped with a locally finite measure ...
2
votes
1answer
94 views

Balls and transformed sets in normed vector spaces

Let $T$ be a surjective, continuous linear operator between two Banach spaces $E$ and $F$. Assume that it is $B_F(y_0,4c)\subset \overline{T(B_E(0,1))}$, where $c>0$, $y_0 \in F$ ($B$ is for ...
2
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0answers
217 views

Question about proof of Stone-Weierstrass

I would like to know if I understand the details in the proof of Stone-Weierstrass (in $\mathbb R$) so I'd like to post it here in my own words. Can you please check it and tell me if it's correct? ...
7
votes
2answers
2k views

Kernel of $T$ is closed iff $T$ is continuous

I know that for a Banach space $X$ and a linear functional $T:X\rightarrow\mathbb{R}$ in its dual $X'$ the following holds: \begin{align}T \text{ is continuous } \iff \text{Ker }T \text{ is ...
3
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2answers
152 views

Proof of the lemma used in proving that a finite-dimensional normed space is complete

I'm trying to understand the proof for the lemma: $$\|\alpha _1 e_1 + \alpha _2 e_2 + \cdots + \alpha_n e_n\| \geq c (|\alpha_1|+|\alpha_2|+\cdots+|\alpha_n|)$$ where $c>0$ and the $e_i$s are ...
1
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1answer
278 views

Question about proof of Arzelà-Ascoli

(Arzelà-Ascoli, $\Longleftarrow$) Let $K$ be a compact metric space. Let $S \subset (C(K), \|\cdot\|_\infty)$ be closed, bounded and equicontinuous. Then $S$ is compact, that is, for a sequence $f_n$ ...
2
votes
1answer
369 views

Completion of $C_c$ with respect to $\|\cdot\|_\Psi$

I'm doing the second half of the following exercise in my lecture notes: "Let $C_c(R)$ be the vector space of continuous functions $f : R \to R$ with $\mathrm{supp}(f)=\overline{ \{x \in R \mid ...
2
votes
1answer
60 views

$a\mapsto \log\left(\lVert f\lVert_{1/a}\right)$ is a convex map

Proof for $a\in (0,1)$ that $a\to F(a)= \log\left(\lVert f\lVert_{1/a}\right)$ is a convex map, if $f\in L^p(X)$ for all $p\geq1$ in some measure space $(X,\mu)$. I'd like to prove that for all ...
3
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0answers
122 views

Question about proof of completeness of $L^p$

In my notes we prove completeness of $L^p$ by showing that if $\sum\|f_k\|_p < \infty$ then $\sum f_k$ converges in $L^p$. (That's a lemma we prove a bit earlier, namely that $(V, \|\cdot\|)$ is ...
6
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2answers
411 views

Closure of the span in a Banach space

Let $X$ be a Banach space, and $S$ a subset. Is it true that $\overline {\operatorname{span}(S)}$ is equal to the set of the elements of $X$ that are obtained as norm convergent infinite sums of the ...
9
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2answers
231 views

If $V \times W$ with the product norm is complete, must $V$ and $W$ be complete?

Let $V,W$ be two normed vector spaces (over a field $K$). Then their product $V \times W$ with the norm $\|(x,y)\| = \|x\|_V + \|y\|_W$ is a normed space. Using this norm it's easy to show that if ...
2
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0answers
110 views

Do the $\ell_p$ spaces have this property?

Let $p\in[1,\infty)$ and consider $\ell_p$. Let $A=\{x=(x_n)\in\ell_p: x_n\ge0\text{ for all }n\in \mathbb{N}\}$. Is there a sequence $u=(u_n)\in\ell_p$ such that $\inf\{x, n\cdot u\}\uparrow x$ as ...
2
votes
1answer
851 views

A compact operator is completely continuous.

I have a question. If $X$ and $Y$ are Banach spaces, we have to prove that a compact linear operator is completely continuous. A mapping $T \colon X \to Y$ is called completely continuous, if it maps ...
8
votes
2answers
733 views

Weak-* sequential compactness and separability

Let $X$ be a Banach space, and let $B$ be the closed unit ball of $X^*$, equipped with the weak-* topology. Alaoglu's theorem says that $B$ is compact. If $X$ is separable, then $B$ is metrizable, ...
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vote
2answers
163 views

How to show $\alpha(A)\leq \beta(A)\leq 2\alpha(A)$

Let $X$ be a metric space and let $A\subset X$ be a bounded subset of $X$. I read on Wikipedia that the Hausdorff- and Kuratowski measures of non-compactness ($\alpha$, resp. $\beta$) satisfy the ...
2
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1answer
91 views

A question about weakening the conditions of Schauder's fixed point theorem

I'm currently doing a course on the theory of metric spaces. This is the version of Schauder fixed point theorem from my course: Let $(X,\|\cdot\|)$ be Banach and $C\subset X$ a closed, bounded, ...
8
votes
1answer
535 views

Inclusion of $L^p$ spaces

Let $X \subset L^1(\mathbb{R})$ a closed linear subspace satisfying \begin{align} X\subset \bigcup_{p>1} L^p(\mathbb{R})\end{align} Show that $X\subset L^{p_0}(\mathbb{R})$ for some ...
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vote
1answer
66 views

Proving $\inf\limits_{f\in\Gamma} \{ F(y(t))(f(t))\}= -\|F(y(t))\| $

I would like to proof the next claim: Let $X$ a Banach space, $F\colon X\to X^*$ a linear continuous function, $$ \Gamma:=\{f\in (\mathcal{C}([0,1],X)\,:\, f(0)=f(1)=0\mbox{ and }\|f\|\leq 1\} $$ ...
8
votes
1answer
136 views

Criterion for convergence of the sequence of powers of a linear operator to $0$

Let $T$ be a linear operator in a Banach space $\mathbf{B}$. Suppose that for every $x \in \mathbf{B}$ there exists some real numbers $c_x>0$ and $a_x<1$ such that $||T^nx|| \leq ca^n$, for all ...
1
vote
0answers
84 views

Does weak*-separability of $X^*$ imply that $X$ is separable? [duplicate]

Possible Duplicate: Does separability follow from weak-* sequential separability of dual space? $\omega^*$-separability of $l_\infty^*$. Recently I read a Theorem stating, Let $X$ be a ...
1
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1answer
71 views

Something weaker than the Riesz basis

I have some function $f$, real valued and continuous. I formed functions $\{f_{m,k}, k \in \mathbb{Z}, m>0\}$ such that that $\mathrm{span}\{f_{m,k}, k \in \mathbb{Z}, m>0\}$ is dense in ...
1
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0answers
75 views

Complementation of $\ell_1$ in dual spaces [duplicate]

Possible Duplicate: Weakly compact operators on $\ell_1$ There is another question I would like to ask, if you don't mind, which is not very far from my previous one. Suppose we have a ...
3
votes
1answer
464 views

Weakly compact operators on $\ell_1$

Is the following assertion true/known? Let $V$ be a Banach space and let $T\colon \ell_1\to V$ be a bounded linear operator. Is it true that $T$ is not weakly compact if and only if there is a ...
1
vote
2answers
400 views

Is an open linear map closed (to some extent)?

Suppose we have a surjective bounded linear operator acting between Banach spaces. By the Open Mapping Theorem it maps open sets in the domain to open sets in the codomain. Must the image of a closed ...
2
votes
1answer
625 views

An application of Riesz' Lemma

How does one prove using Riesz' Lemma that an infinite dimensional subspace $Y$ of a Banach space $X$ contains a sequence $\{x_n:n\in \mathbb{N}\}$ in the unit ball of $Y$ such that $n \neq m$ implies ...
2
votes
2answers
294 views

compact projections to infinite dimensional Banach spaces

If I consider $X$ to be an infinite dimensional Banach space and $P\in P(X)$, that is, $P$ is a continuous linear projection. How does one prove that $P$ is compact if and only if $\dim R(P)$ is ...
1
vote
1answer
176 views

Is the range of this operator closed?

I think I am stuck with showing closedness of the range of a given operator. Given a sequence $(X_n)$ of closed subspaces of a Banach space $X$. Define $Y=(\oplus_n X_n)_{\ell_2}$ and set $T\colon ...