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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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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50 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
30 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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83 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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5answers
266 views

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 $\lbrace x_n \rbrace$ ...
5
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1answer
41 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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52 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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90 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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21 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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80 views

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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45 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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41 views

Weak topology in Banach spaces

Let $X$ be a Banach space and suppose that $\{x_n\}$ is a sequence in X that is bounded, and suppose that $f(x_n) \longrightarrow f(x)$ $\forall f \in A$ where $A$ is subset dense of $X^*$. Show ...
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1answer
15 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
22 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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18 views

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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1answer
49 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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64 views

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 ...
2
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1answer
48 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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32 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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33 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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59 views

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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1answer
45 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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41 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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2answers
22 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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22 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
25 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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53 views

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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26 views

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
23 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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24 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 ...
3
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1answer
46 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
53 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
16 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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78 views

$L^{p}$ 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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49 views

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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30 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
18 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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23 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
19 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
70 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
99 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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1answer
38 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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29 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 ...
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1answer
27 views

Nonequivalent norms on infinite-dimensional linear space

I've just proven that for every infinite-dimensional space with a norm $(V, ||~||)$ we can find a discontinuous linear functional $ \phi $. But next I'm trying to prove the following: The norm $ ...
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48 views

Space of real polynomials of one variable isn't complete

Consider $E$ - the vector space of all real polynomials of one variable. I need to prove that it is not complete under any norm. I was thinking I could use the fact that certain functions, for ...
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1answer
136 views

Why isn't $\,\mathcal C[0,1]$ a Banach space in this unusual norm?

I wish to ask the following question: Let $\mathcal X$ be the normed space $\,\mathcal X=\mathcal C([0,1])$, with norm defined as $$ \|\,f\|= \max_{x\in[0,1]} x^2 \lvert\,f(x)\rvert. $$ Why isn't ...
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41 views

Separable dual space implies existence of complete series

Let $\{x_n\}$ be a basis for a Banach space $X$ and let $\{f_n\}$ be the associated sequence of coefficient functionals. Prove or disprove: if $X^\ast$ is separable, then ${f_n}$ is complete ...
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112 views

Is $\mathcal{B}(H)$ complemented in $\ell_\infty(I, H)$

Let $H$ be an infinite diensional Hilbert space. Consider unit ball of $H$ as index set, denote it by $I$, then we have an isometric embedding $$ ...
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1answer
34 views

Neighborhood base of weak topology

If $X$ is a Banach space, the weak topology on $X$ is the weakest topology in which each functional $f$ in $X^\ast$ is continuous. I have some difficulties in understanding its neighborhood basis in ...