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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Is the space of continuously differentiable functions over Polish spaces Polish?

Let $(X, \|\cdot\|_X)$ and $(Y, \|\cdot\|_Y)$ be two separable Banach spaces. Consider the space of continuously differentiable functions mapping $X$ to $Y$; i.e. $C^1(X, Y)$. Consider the usual ...
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w*-convergence vs. convergence on a dense subspace

Let us have a Banach space $X$, a dense subspace $D\subseteq X$, a net $\{\phi_{i}\colon i\in\mathcal I\}$ in $X^*$ and $\phi\in X^*$. Suppose that $$\lim\limits_{i\in\mathcal I}\phi_{i}(d)=\phi(d)$$ ...
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73 views

Normed linear space with two norms that are not equivalent, one is complete, what about the other?

I have been searching for an answer to the following question: Given a normed linear space $V$ and two norms that are not equivalent, but $\exists K\in\mathbf{R}$ such that $\|v\|_1\leq K\|v\|_2$ ...
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77 views

Closure of compact sets in Banach space

Let $(X,\vert\vert\cdot\vert\vert)$ be a Banach space. For each $k\in\mathbb{N}$ let $A_k\subseteq X$ be compact and $r_k\in\mathbb{R},r_k>0$, such that $$A_{k+1}\subseteq \{x+u\vert x\in A_k ...
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Dual of Hilbert space dense in dual of Reflexive space.

I don't see how to solve this problem which I think should be easy: Let Y be a reflexive space. Assume $Y$ is continuously embedded in a Hilbert space $H$ and $Y$ is dense in $H$. Show that $H^*$ is ...
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1answer
62 views

Uncompletion of a Banach Space

Given any Banach space there is a way to define a norm such that is no longer complete. I know you can reach the result by using a Hamel base $H$ for this given space and doing this: If $$ \forall ...
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The Haar basis ,proof of orthonoramality.

please i have this problem and i known how to prove completeness but do not know how to prove that it is orthonormal. I will appreciate it if anyone can help me. Given that $n\geq1$ write ...
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A useful criterion in dispersive PDE.

I would like to prove the following theorem: Consider two Banach spaces $X\hookrightarrow Y$ and $1<p,q\leq\infty$. Let $(f_n)_{n\geq 0}$ be a bounded sequence in $L^q(I,Y)$ and let $f:I\mapsto Y$ ...
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closed unit ball in a Banach space is closed in the weak topology

Let $V$ be a Banach space. Show that the closed unit ball in $V$ is also closed in the weak topology. I know this is a consequence of the statement any closed convex subset in $V$ is closed in the ...
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23 views

understanding a proof involving equivalence of norms in finite dim. linear normed spaces

I am reading the proof of the theorem shown below (from Linear Functional Analysis by Rynne and Youngson). I can't figure out why the part I highlighted in red is true. I understand why $S$ is compact ...
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54 views

Banach space problem

I came across the following problem: For an open set $U$ in $\mathbb{R^n}$ we define the set of all k-times continuously differentiable functions $f:U\rightarrow \mathbb{R}$ for which $D^\alpha f$ ...
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Is there anything wrong with this line of reasoning?

Proof statement: If $f:X \rightarrow Y$ is a bounded linear map, then $Df(x)=f$ for all $x \in X$, where $X$ and $Y$ are Banach spaces. Proof: Consider $$\lim_{h\rightarrow ...
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1answer
34 views

Bounded operators on unit ball and equicontinuity

Let $X$ be a Banach space and $B$ be the closed unit ball contained in $X$. Let $\{T_{\alpha}\}$ be a family of bounded linear operators from $B$ to $V$, a normed vector space. My question is: Suppose ...
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1answer
81 views

Completeness and separability of $B(X)$ (bounded linear operators on $X$)

Assume $X$ is a separable Banach space with norm $|| \cdot||$. Consider $\{f_n\}_{n \in N}$ a countable dense subset of $X$ and equip $B(X)$ (bounded linear operators on $X$) with the following ...
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2answers
62 views

Has this theorem (on existence of inverse) an analogous for unbounded operators?

Let $S,T:X\to X$ be bounded linear operators, where $X$ is a Banach space. It's a consequence of the Banach Fixed Point Theorem that if $T$ is invertible and $\|T-S\|\|T^{-1}\|<1$ than $S$ is ...
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Largest/smallest cross-norm: A simple question about cross-norms on tensor products of Banach spaces.

This is a very simple dumb question as I’m completely new to the topic. I was reading Wikipedia’s entry on “Topological Tensor Products”, and there’s one thing I’m confused about. Let $ A $ and $ B $ ...
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40 views

Counterexample for the Chain rule for the Gateaux-derivative

I'm reading the book of Drabek, Milota - Methods of Nonlinear Analysis, and at page 121, they state: but I can't manage to find such counterexample. For clarity the Gateaux derivative is defined ...
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105 views

Confused about a version of Schauder's fixed point theorem

I have read this: We have a map $S:W_0 \to W_0$. Moreover $W_0$ is not empty, convex, and weakly compact in $W$. Thus we can apply Schauder's fixed point theorem: Schauder's fixed point ...
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36 views

Understanding the right assumption about Gateaux derivative in Banach spaces

It is a general fact that the notion of Gateaux derivative is not uniform over the mathematical community i.e. someone requires it to be linear and continue other not and require this additional ...
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2answers
56 views

a functional analysis question

$X$ is a banach space and $f$ a non zero linear functional. I'm trying to show $null(f)$ not dense in X $\implies f$ continuous. I've tried a few approaches but I think the following seems the most ...
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25 views

Is the set, $\{f\in Y: \left\|f\right\|_{Y} \leq C \ \text{and} \ \left\|(|f|)\right\|_{X}\leq C \}$, closed in $(Y, ||\cdot||_{Y})$?

Put, $X= \text{The space of "nice" complex valued functions on } \mathbb R; \text{that is}, f:\mathbb R \to \mathbb C;$ so that $X$ is Banach space with respect to the norm ...
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1answer
55 views

Invariant convex subset

Let X be a banach space and $T$ a bounded operator on X with $||T||$ less than or equal to 1. If $T$ is an isometry and $r(T)$<1 show that there is a closed convex non-zero proper subset $C$ of the ...
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63 views

Duality mappings on finite-dimensional spaces

I have a few questions regarding some concepts in a book "nonlinear partial differential equations by Roubicek" I am studying. The following is from the text. "Let $V$ be a separable, reflexive ...
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34 views

when one can expect $\left \| |x|^{2}x- |y|^{2}y \right \| \leq \frac{1}{2} \left \| (x-y) \right \|$?

Suppose $(X, \left\|\cdot \right \|)$ is Banach algebra with the property that $\left\| (|x|^{2}x) \right\| \leq \left\|x\right \|^{3};$ for every $x\in X.$ Let $y_{0}\neq 0 \in X$ and fix it; and ...
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26 views

infinite wedge product

In the theory of manifolds, we use wedges to help define integration and after some work we prove Stoke's theorem. I'm wondering if we can make sense of integration for a Banach manifold (locally each ...
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43 views

A subspace isomorphic to $C[0,\alpha^{\omega}]$.

Let $\omega\leq \alpha<\omega_1$ ordinal, $Y$ Banach space, then $\ell_{\infty}^*\otimes_{\pi}Y$ has a subspace isomorphic to $C([0,\alpha^{\omega}])$? For me it would be nice if the answer was no. ...
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37 views

Bochner measurability

I have the following problem. Let $(\Sigma, \Omega, \mu)$ be a measure space and let $X$ be a Banach space. Take a function $f \colon \Omega \rightarrow \mathbf{B}(X)$, which takes values in space of ...
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1answer
27 views

Proving an operator $D: L^2[0,1]\rightarrow C'$, $Df(t)=\int^t_0 f(s) ds$ is unitary

Let $C'\subseteq C[0,1]$ be the space of all absolutely continuous function such that $f(0)=0$ and $f' \in L^2[0,1]$. Define an inner product on $C'$ as $\langle f,g \rangle = \int^1_0 f'(t)g'(t)dt$. ...
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Abstract Wiener space and integration related to a trace class operator

Suppose I have a trace class operator $A$ of a Hilbert space $H$. Also suppose I have an abstract Wiener space $(H,B)$. Then, $\langle Ax, x \rangle$ is defined almost everywhere in $B$ with respect ...
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Subspaces in the image of compact operator

Let $X$ and $Y$ be some infinite dimensional Banach spaces. Let $T:X\longrightarrow Y$ be some compact linear operator. It is easy to understand that $T$ cannot be surjective: the Open Mapping Theorem ...
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1answer
50 views

Are $C^0[a,b]$ and $C^0[0,1]$ isometrically isomorphic?

Consider $C^0[a,b]$ and $C^0[0,1]$, each equipped with the $L^1$-Norm. Are these (out of curiosity) isometrically isomorphic?
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Is there example for isomorphic closed subspaces of a Banach space with non isomorphic quotient?

$Y_1$ and $Y_2$ are closed subspaces of a Banach space X and $Y_1 \simeq Y_2$. I can't find a way to show $X/Y_1 \simeq X/Y_2$ and it made be think that it's not true. Is there a counter example?
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Show the Banach-Mazur distance is reached for finite-dimensional Banach spaces

Let $X,Y$ be isomorphic Banach spaces. The Banach-Mazur distance: $d(X,Y)=\inf \{\| T\| \| T^{-1}\| : T:X\rightarrow Y \text{is an isomorphism} \}$ can be rewritten as: $d(X,Y)=\inf \{\| T^{-1}\| ...
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Bounded inverse theorem.

http://planetmath.org/boundedinversetheorem referring to this proof i don't get the final statement: "$T^{-1}$ is continuous, i.e. bounded". I know that the boundness would be surely true if $T^{-1}$ ...
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66 views

When are two Banach spaces equal?

Suppose $X$ and $Y$ are two Banach spaces. If there is an isometric isomorphism, then we can say that these two spaces are same. Is there any other condition which will say that these two spaces are ...
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Question about perpendicular complements in Banach spaces

Let $V,W$ be Banach spaces with $T : V \to W$ a bounded linear transformation. Let $T^* : W^* \to V^*$ be the standard adjoint map on the dual spaces. That is, for $g \in W^*$, we have $[T^*g](v) = ...
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A Banach space that is not a Hilbert space

Can someone give me an example of a Banach space that is not a Hilbert space? I can't think of any because I don't know how to show one space that can not have inner product structure.
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Why do we need “closed” here?

There is a statement such that: Every closed finite co-dimensional subspace of a Banach space is complemented. I don't really see why we need the subspace to be closed. If $X$ is a Banach space and ...
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1answer
54 views

Non-existence of minimisers in $L_1$ and $L_{\infty}$

In this note, Exercise 11 asks for finding counterexamples to the existence of minimisers in $L_1$ and $L_{\infty}$, which is For $p = 1$ or $=\infty$, there exists a Banach space $L_p(X,\mu)$ ...
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74 views

Uniform boundedness principle for norm convergence

This is from Tao's book; Let $X$ be Banach space, let Y be normed vector space, and let $(T_n)_{n=1}^{\infty}$ be a family of continuous linear operators $T_n : X \to Y$. Then following is ...
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2 simple questions about Banach space

Given that X is a normed space. Let $S=\left \{ x \in X : \left \| x \right \|=1 \right \}$. Prove that $X$ is a Banach space iff every Cauchy sequence $\left\{x_{n}\right\}$ in $S$, there is $x$ ...
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24 views

Summability of Fourier series from Banach space point of view

I am under the impression the following is true (any pointer to a reference would be appreciated ): Theorem (Katznelson?) For any $f \in C[0,1]$ with Fourier coefficients $\{ \hat{f}(n)\}$, there ...
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1answer
63 views

Proposed proofs for weak convergence question

I have the following question and two proposed proofs. Please advise if these proofs are adequate and which of the two is better. Thanks. Question: Let $V$ be a reflexive, separable Banach space. ...
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31 views

The equivalent definition of denting point

How i can prove that If $K$ is a subspace of Banach space $X$, $x$ is denting point of $K$,when for every $\varepsilon>0$,there is a unit vector $x^{*}\in X^{*}$ and $\delta>0$ such that ...
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1answer
16 views

SOT Convergence and Compact Convergence

Let $E$ be a Banach space, and let $A(E)$ denote the closure of the finite rank opertors on $E$. Let $(S_\alpha)$ be a bounded net of operators on $E$ such that $S_\alpha T\to T$ for all $T\in A(E)$. ...
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1answer
38 views

Proof check for completeness

I'd like to see if the proof I have is adequate. Statement. Let $X$ and $Y$ be Banach space, the product $X\times Y$ is a vector space under coordinate operations with norm $$ \|(x,y)\| = \|x\|_X ...
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1answer
69 views

If every closed subspace of a Banach space has a closed orthogonal complement, then it is a Hilbert space.

My professor mentioned this fact in class. FACT: If every closed subspace of a Banach space has a closed orthogonal complement, then it is a Hilbert space. He mentioned that he had never seen the ...
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36 views

Are all functions in a Banach space convergent?

Are all functions in a Banach space convergent? I need this answer in a study of wavelet analysis. My thoughts are: since we have this definition: Let $X$ be a Banach space. A sequence of vectors ...
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1answer
66 views

This is Banach space?

Let me denote by $C_{0}(\mathbb{R})$ the set of continuous functions which tend to zero at + and - $\infty$. I am wondering if it is true that $(C_{0}(\mathbb{R}), \Vert . \Vert_{\infty})$ is a ...
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116 views

Density of $C^{1}_{0}(\mathbb R)$ in $L^{\infty}(\mathbb R)$

I am looking for a counterexample to $C^{1}_{0}(\mathbb R)$ ( $C^1$ functions with compact support) is dense in $L^{\infty}(\mathbb R)$? Is there some easy counterexample showing that this latter is ...