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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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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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). ...
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401 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 ...
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1answer
151 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
185 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 ...
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1answer
285 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
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 ...
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1answer
175 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 ...
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3answers
190 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\| + ...
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2answers
291 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 ...
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135 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 ...
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146 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 ...
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2answers
197 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?
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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 ...
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276 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 ...
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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 ...
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219 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? ...
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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 ...
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2answers
153 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 ...
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1answer
280 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 $f_n$ ...
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1answer
376 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 ...
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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 ...
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125 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 ...
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417 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 ...
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232 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 ...
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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 ...
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1answer
867 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 ...
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742 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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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 ...
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1answer
92 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, ...
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1answer
538 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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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, $$ \Gamma:=\{f\in (\mathcal{C}([0,1],X)\,:\, f(0)=f(1)=0\mbox{ and }\|f\|\leq 1\} $$ ...
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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 ...
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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 ...
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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 ...
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80 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 ...
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1answer
469 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 ...
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2answers
411 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 ...
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1answer
630 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 ...
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299 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 ...
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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 ...
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1answer
63 views

An explicit example of an invariant halfspace of the unilateral shift?

In a recent talk, A. Popov stated the following fact The unilateral shift on $\ell^2$ has invariant halfspaces. Halfspaces are closed subspaces whose dimension and codimension are both infinite. ...
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166 views

The separation theorem for the w* topology

When I read the prove of the Goldstine Theorem(See An Introduction to Banach Space Theory Robert E. Megginson 2.6.26), I find it use the separation theorem for the w* topology without any details. ...
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1answer
60 views

Convergent operatorial series

An exercise I was doing asks (among other things) for the values of $z\in\mathbb{C}$ for which the following (operatorial) series converges absolutely: $$\sum_{n=0}^{\infty}z^nA^n$$ where $A$ is an ...
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1answer
159 views

Banach space, Normed vector space

Help me please with this question. Let's $Y$ be Banach space, $Z$ - Normed vector space and $(T_{n})_{\mathbb{N}}$ - the sequence in $B(Y,Z)$ so that all sequence $(y_{n})_{\mathbb{N}}$ in Y holds: ...
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179 views

Dual without quotients isomorphic to $c_0$

The following question is bothering me. Suppose we have a dual Banach space $X^*$ and assume $X^*$ has a quotient isomorphic to $c_0$. Must $X^*$ contain a complemented copy of $\ell_1$?
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247 views

Criterion for a limit of invertible operators on a Banach space to be invertible

Let $A_n$ linear operators in a Banach space $B$ that have inverses. $||A_n-A|| \to 0$ for some operator $A$. I need to prove that $A$ has an inverse operator iff the sequence $\{||A_n^{-1}||\}$ is ...
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193 views

Embedding dual of a Banach space into a predual

Let $X$ and $Y$ be Banach spaces and suppose moreover that there is an isometric embedding of $X^{**}$ into $Y$. Assume moreover that $Y$ has the unique predual $Y_*$ up to isometry (like von Neumann ...
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729 views

Must a weakly or weak-* convergent net be eventually bounded?

Let $\mathfrak{X}$ be a Banach space. As a standard corollary of the Principle of Uniform Boundedness, any weak-* convergent sequence in $\mathfrak{X}^*$ must be (norm) bounded. A weak-* convergent ...
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1answer
175 views

An upper bound for $\|(\lambda-A)^{-1}\|$?

Let $A$ be a k-by-k matrix and $\sigma(A)$ its spectrum, or the collection of eigenvalues of $A$. If we know $\lambda\notin\sigma(A)$, then $\lambda$ is at a positive distance to all points in the ...