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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Understanding dual spaces

My professor asked me to give him an example about dual spaces from our real life so I was thinking if we take the set of all our actions as the space X can the space of all our thoughts be the dual ...
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Proof Riesz Representation Theorem (bounded linear functional in Lp)

I have a little problem with this proof (I'm using Royden), can you help me? Let $F$ be a bounded linear functional on $L^p$, $1 \leqslant p \leqslant \infty$. Then there is a function $ge \in L^q$ ...
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65 views

Weak Convergence and Weak Topology

In discussing weak topology of a normed space $X$, a lemma is given as follows. If $(x_n)$ is a sequence in $X$ converging weakly to $x$, then $x_n$ is bounded. I understand the proof of this ...
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25 views

Continuous Strong-Strong Implies Continuous Weak-Weak

Let $X$ and $Y$ be two Banach spaces and let $T$ be a linear map between $X$ and $Y$. Show that $T$ is continuous strong-strong if and only if $T$ is continuous weak-weak. I can see that $T$ ...
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Dual of a locally convex space

Let $X$ be a normed space. Suppose $E$ is a subset of $ X^*$ (The space of continuous linear functionals). For every $\phi\in E$, define seminorm $p_\phi: X\to [0,\infty)$ such that $p_\infty (x)= ...
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123 views

Banach Spaces: Uniform Integral vs. Riemann Integral

Reference For a bounded nonexample of integrability see: Riemann Integral: Bounded Nonexample For a convergence theorem on integral see: Riemann Integral: Uniform Convergence For an improper ...
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50 views

If $f(x)$ is close to $0$ then necessarily $x$ is close to $\ker f$?

Suppose that $X$ is a real Banach space and $f:X \to \mathbb{R}$ is a continuous linear functional. Is it true that for any $\varepsilon>0$ there is a $\delta>0$ such that for any $x \in X$ we ...
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59 views

A Question on Compact Operators on Hilbert Space

I read this question which I have no idea how to start. Could anyone provide me with some detailed answer, please? Thanks. Suppose that a linear operator $F$ from a Hilbert space $\mathcal H$ to ...
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1answer
38 views

Inner products of weakly convergent sequences

I have a weakly convergent sequence in $L^2(U)$ (for $U$ some bounded open domain with smooth boundary), $u_k\rightharpoonup u$. I want to show that there is a sequence $v_k\rightharpoonup v$, such ...
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1answer
85 views

Best approximation for a closed set in a finite dimensional normed space

First of all I'd like to mention that it is a part of my home work so I'd like if you won't give the answer itself, but try to guide me into it. I've been losing my mind for the last couple of hours ...
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Possible flaw in “proof” that a sum of two compact operators is compact

If X and Y are Banach spaces, and $A: X \to Y$, $B: X \to Y$ are both compact operators, then $A + B$ is compact. A + B is compact if and only if for every bounded sequence $\lbrace x_n \rbrace$ ...
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45 views

limit of a weakly convergent sequence in Banach spaces

Consider $X$ a Banach space. For some sequence $x_n\in X$, assume that for every $f\in X^*$, $f(x_n)\to c_f$. Does this imply that $\exists x\in X$ where $c_f = f(x)$? How might I go about finding ...
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2answers
59 views

Property of linear projections

Theorem 15 in Chapter 15 of Peter Lax's functional analysis book says $X$ is a Banach space, $Y$ and $Z$ are closed subspaces of $X$ that complement each other $X = Y \oplus Z$, in the sense that ...
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97 views

A smooth nonzero function $\mathbb R\to\mathbb R$ with uniformly bounded derivatives tending to zero at infinity?

$\def\scrBp{\mathscr B\boldsymbol.}\def\rD{{\rm D}\kern.4mm}\def\ssp{\kern.4mm}\def\bbR{\mathbb R} $For a certain purpose I invented the Banach space $\scrBp^\infty(\ssp\bbR\ssp)$ defined as follows. ...
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23 views

Is $C^0_c(\Omega)$ complete wtr the $L^2(\Omega)$ norm?

$\Omega$ is an open convex subset of $\mathbb R^N$ I think it isn't. To prove this I show that $C^0_c(\Omega)$ isn't closed: there is a sequence in $C^0_c(\Omega)$ with limit (wtr the $L^2$ norm) not ...
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How do endomorphisms of Banach space modulo compact operators look like?

It is well-known that given a Banach space $X$, the set of compact operators (let's denote it by $K(X)$) on $X$ forms a both-sided ideal in $L(X)$, the ring of bounded linear operators on $X$. My ...
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2answers
53 views

Determination of some operator norms

I have to determine the operator norms, the kernels and the images of the following 2 maps: 1) $F_1 :\{x\in C^0([0,10],\mathbb R)|x(0)=0\}\rightarrow C^0([0,10],\mathbb R)$ ...
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1answer
16 views

Nonclosable Operator: Example (Wikipedia)

The example here is taken from the wikipedia article: Discontinuous Linear Map Given the spaces of polynomials $X:=\mathcal{P}([0,1])$ and $Y:=\mathcal{P}([2,3])$. Their completions being ...
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1answer
53 views

find convergent subnet

I`m trying to learn how to use nets and subnets to prove some theorems. I came across this problem in Banach space book: let K be a compact subset of a topological space X. suppose that ...
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1answer
26 views

Showing space of functions with lipschitz norm is complete

I have a Banach space, $X$, given by all the complex valued functions $x: [-1,1] \to \mathbb{C}$ where $x(0) = 0$. And I've shown that the following defines a norm on $X$: $$\|x\| = inf \{ \beta : ...
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Finding a norm on $ \mathbb{R}^X $ such that the “natural” embedding of a metric space $ X $ in $ \mathbb{R}^X $ becomes an isometry

This question is related to this one: Every metric space can be isometrically embedded in a Banach space, so that it's a linearly independent set Kuratowski's embedding indeed seems to be the ...
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92 views

Existence of a mapping in a nonseparable Banach space that moves all nearby points to far-away points

Does there exist a nonseparable Banach space $X$, a mapping $F: X\to X$, and an open nonempty subset $D\subset X$ such that $$ \forall\,E>0 \quad \exists\,\delta>0: \quad \forall\,x,y\in D \quad ...
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53 views

Proof about distances in banach spaces

Let $(X,\|\circ\|)$ be a banach space with $$\forall x,y\in X, x\neq y, \|x\|=\|y\|=1 \Rightarrow \|\frac{x+y}{2}\|<1 $$ If M is convex and $z \in X$, then $\|x-z\|=dist(z,M)$ for at most one $x ...
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James $\ell_1$-theorem: problem in the proof

I am struggling with the very last estimate in the proof of James' $\ell_1$-theorem. (Please see below an excerpt from Albiac and Kalton's fantastic book Topics in Banach space theory.) I don't ...
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1answer
61 views

Every metric space can be isometrically embedded in a Banach space, so that it's a linearly independent set

I'm trying to prove the following: Every metric space can be isometrically embedded in a Banach space, so that it's a linearly independent set. I came up with the following idea: Let $ (X,d) $ be a ...
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34 views

Banach spaces, partial sums converge? [closed]

Let $(X,||.||)$ be a Banach space. Suppose the sequence $(x_n) \subset X$ is such that $\sum_{n=1}^{\infty} \|x_n\|$ converges in $\mathbb{R}$. Prove that $S_n = x_1+x_2+...+x_n$ the "Partial sums" ...
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37 views

Definition of reflexive Banach spaces

I'm trying to understand the definition of reflexive spaces. I wrote in my notes: If $Y$ is reflexive then for all $\eta\in Y^{**}$, $f\in Y^*$, $\exists y\in Y$ where $\eta(f) = f(y)$. My question ...
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About the adjoint operator and weak operator topology.

Let $X,Y$ be Banach spaces. Let $\lbrace{S_n\rbrace}\subset\mathcal{L}(X,Y)$, and $T\in\mathcal{L}(X,Y)$, such that $S_n\xrightarrow[n\to\infty]{WOT}T$, that is: $$\langle ...
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72 views

A Counter Example about Closed Graph Theorem

This is an example I read around closed graph theorem. Let $Y=C[0, 1]$ and $X$ be its subset $C^\infty[0, 1]$. Equip both with uniform norm. Define $D: X \to Y$ by $f \mapsto f'$. Suppose $(f_n, f_n') ...
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52 views

Weak convergence and convergence in distribution

Is convergence in distribution related to weak convergence in Banach theory? Where by weak convergence I mean: for every functional f the sequence $\langle f,x_n\rangle \overset{n}{\rightarrow} ...
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28 views

Completeness of the space of random variables with bounded conditional first moment with respect t0 $\left\Vert \cdot\right\Vert _{2} $

Consider a probability space $\left(\Omega,\mathcal{F},P\right) $, and a sub-sigma-algebra $\mathcal{G}\subseteq\mathcal{F} $. As usual, let $L^{2}\left(\Omega,\mathcal{F},P\right) $ be the space ...
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24 views

Soft: Interesting examples of applications of the Itô–Nisio theorem

I'm writing up a presentation on the Itô–Nisio theorem and I'm looking for a simple (nontrivial) example showcasing it. I was thinking of a 2-dimensional simple random walk, but that seems ...
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1answer
30 views

Proof completeness of $L^p$

I'd like to check if I understood the proof that $L^p$ is complete ($1 \le p <+\infty$). I have to use the following fact: in a metric space, if a Cauchy sequence has a convergent subsequence then ...
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Why $L^1$ is not reflexive [duplicate]

We already known that $$ (L^p(\Omega))^* = L^q(\Omega), $$ for all $1\le p < \infty $ and $q$ is the exponent conjugate to $p$. So that, $L^p(\Omega)$ is reflexive with $1<p<\infty$. However, ...
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Given $X,Y$ are Banach spaces with norms $\|x\|_X,\|y\|_Y$, prove $\|(x,y)\|=\max(\|x\|_X,\|y\|_Y)$ is a norm and defines a Banach space

Here is the question: Let $X$ and $Y$ be Banach spaces with norms $\|x\|_X$ and $\|y\|_Y$ respectively. Prove that $$\|(x,y)\|=\max\{\|x\|_X,\|y\|_Y\}$$ defines a norm on $X\times Y$, and that ...
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1answer
27 views

Find $f$ such that the contraction $\phi$ has a fixed-point $\rho= \sqrt{2}$

I use the Newton method and the Banach fixed-point theorem and have: Let $I \subset \mathbb{R}$ a closed interval and $\phi: I \rightarrow I$ Lipschitz continous and $f: I \rightarrow \mathbb{R}$ a ...
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32 views

Banach fixed-point theorem and Newton

I have to combine the Newton method and the Banach fixed-point theorem: Let $I \subset \mathbb{R}$ a closed interval and $\phi: I \rightarrow I$ Lipschitz continous. Let $f: I \rightarrow ...
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62 views

Holder Inequality

I have a problem with the demonstration of this inequality ('m following Royden) : If $p$ and $q$ are nonnegative numbers such that $\frac{1}{p}+\frac{1}{q}$ and if $f \in L^p$ and $g \in L^q$, then ...
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1answer
57 views

$p$-summable series in a Banach space

Let $E$ be a Banach space and denote its dual space by $E^*$. Let $p \in [1, \infty)$ and $x : \mathbb{N}\rightarrow E$ be such that for every $\phi \in E^*$, $$\left( \sum_{n=1}^{\infty} \lvert ...
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1answer
26 views

Convergence with respect to the graph norm

Let $T:A\subset E\to F$ be a linear closed operator with dense domain. (E,F are Banach spaces) and $x_n$ a sequence in A such that $||x_n-x||_E\to 0$. Let $||x||_1:=||x||_E+||T(x)||_F$ be the graph ...
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$L^{p}$ spaces and their properties

I have a question: I don't know how to show that if $1<p<q<\infty$ , then $L^{q}(0,1)\subset L^{p}(0,1)$ and $\mid\mid f\mid\mid_p$ < $\mid\mid f\mid\mid_q$, $\,f \in L^{q}(0,1)$?
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Does this function have a unique fixed point?

Consider the following equations in fixed-point form: $$\mathbf{x}=F(\mathbf{x})=[f_1(\mathbf{x}),f_2(\mathbf{x}),\cdots,f_n(\mathbf{x})]^T$$ where the function $f_i$ is defined as ...
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40 views

Newton method and the Banach fixed-point theorem

I try to combine the Newton method and the Banach fixed-point theorem but I have still some questions: Let $I \subset \mathbb{R}$ a closed interval and $\phi: I \rightarrow I$ Lipschitz continous. ...
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1answer
24 views

Isometric Operators: Common Core

Given a Hilbert or Banach space $\mathcal{H}$. Consider two closed operators $S:\mathcal{D}(S)\to\mathcal{H}$ and $T:\mathcal{D}(T)\to\mathcal{H}$. Suppose they're isometric on a common core ...
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24 views

Natural *-isomorphism

Show that if $X,Y$ are locally compact spaces, then there is a natural *-isomorphism from $C_0(X,C_0(Y))$ onto $C_0(X\times Y)$. My attempt: I define $\phi:C_0(X,C_0(Y))\to C_0(X\times Y)$ such that ...
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1answer
25 views

Gelfand triple: what happens if we don't identify the pivot Hilbert space with its dual?

People usually say $V \subset H = H^* \subset V^*$ is a Gelfand triple if $V$ is continuously and densely embedded in $H$ and $H$ is identified with its dual. Sometimes they do not mentioned that ...
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1answer
80 views

Banach fixed point theorem (application)

Let $X$ and $Y$ be Banach Spaces, and $T: X \to Y$, where T is continuous, linear and bijective, and let $S: X \to Y$ (where $S$ is continuous and linear) with $|S|\cdot|T^{-1}|< 1$. Show that ...
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1answer
114 views

What's Helly's theorem in the proof of the Goldstine–Weston density theorem

I have a problem in understanding a proof of Goldstine–Weston density theorem. The only thing I don't know in the proof is the part of Helly's theorem to be related. The Goldstine–Weston density ...
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45 views

Positive linear functional on an involutive Banach algebra

Why is every positive linear functional on an involutive Banach algebra with a bounded approximate continuous?
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30 views

Convergence in $L^1 \cap L^2$

I am very confused about the following: Assume we have a sequence of functions $f_n \in$$L^1 \cap L^2 (\mathbb{R}^n)$. Then is it true that if this sequence is Cauchy both in $L^1$ and $L^2$, two ...