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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Topology induced bycone metric

Is cone metric define atopology as same as the topology define by ametric? I have tried to prove it by theorems that joined them
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How to show this Sobolev space is a uniformly convex space?

From the book by Kufner: How do I prove this theorem? I'd like to do it using the epsilon delta definition (see http://en.wikipedia.org/wiki/Uniformly_convex_space) if possible.
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Proving norm equivalence in $W^{1-1/p,p}(\Omega)$

Define for $p\in [1,\infty)$ and $\Omega=(0,1)^N\subset\mathbb{R}^N$ $$W^{1-1/p,p}(\Omega)=\left\{u\in L^p(\Omega): \ \int_\Omega\int_\Omega\frac{|u(x)-u(y)|^p}{|x-y|^{N-1+p}}dxdy<\infty\right\}$$ ...
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Representation of subspaces as complemented subspaces

Let $X$ be any separable Banach space. The Banach-Mazur theorem states (astonishingly) that $X$ is isometric to a closed subspace of $C(\Delta)$, the space of continuous functions on the cantor set ...
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186 views

Proving that $X/M$ is a Banach space when $X$ is

I am trying to do an exercise in an introduction to functional analysis course: 1) Let $X$ be a normed space and $\{x_{n}\}_{n=1}^{\infty}\subseteq X$. Prove that $X$ is a banach space iff ...
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38 views

if $P\Rightarrow Q$ Then both are banach space?

$X,Y$ are norm linear space and $T_n$ be a sequence of bounded linear operators from $X\to Y$ consider the two statements below $P:\{\|T_n(x)\|\}$ is bounded for ever $n$ $Q:\{\|T_n\|\}$ is bounded ...
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1answer
32 views

Clarification about the space $C^0 ([-T,T],B)$

Let $B$ be a Banach space (in particular, $B$ is a function space equipped with the supremum norm). The space $C^0 ([-T,T],B)$ is the set of continuous functions on $[-T,T]$, valued in $B$. ...
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72 views

$B$ be an Infinite dimensional Banach Space and $T:B\to B$ be an continuos operator

Let $B$ be an Infinite dimensional Banach Space and $T:B\to B$ be an continuous operator such that $T(B)=B$ and $T(x)=0\Rightarrow x=0$ which of the following is correct? $T$ maps bounded sets into ...
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174 views

Complemented Banach spaces.

Let $X$ be Banach space and $Y$ a closed subspace of $X$. Assume that there exist a closed "subset" $Z$ of $X$ with the properties: $Z\cap Y=\{0\}$ and every $x\in X$ can be written in a unique ...
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54 views

characterization of an infinite matrix mapping and continuity

Show that an infinite matrix mapping $A=[a_{ij}]$ $:l^{\infty}\to l^{\infty}$ is continuous iff $sup_{i\in \mathbb N}$ $ \sum_{k=1}^{\infty}{|a_{ij}|}=||A||<\infty$. Give a characterization of the ...
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34 views

proving that bs is banach

Let's define $B_s$ as the of real valued sequences $(x_n)$, such that $sup_{N\in \mathbb N} |\sum_{k=0}^{N}{x_k}| $ is bounded, and make it a vector space considering the usual pointwise operations ...
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366 views

Are these Banach spaces separable?

Let us consider the set $c$ of convergent sequences, and the subspace $c_0$ of convergent sequences to zero. They are Banach spaces over $\mathbb C$ or $\mathbb R $ under the sup-norm (and the usual ...
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552 views

Linear combinations of delta measures

Let us consider the space of Borel, regular, complex measures on the real line, endowed with the total variation norm. Inside this space, I would like to characterize the space of all the finite ...
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1answer
237 views

Exercise in Folland on extending a closed subspace of a Banach space

Folland gives the following problem on page $159$ of his book Real Analysis: Let $\mathcal X$ be a normed space. If $\mathcal M$ is a closed subspace and $x\in \mathcal X \setminus \mathcal ...
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66 views

Is space of Dirac measures Banach?

Is the space of all Dirac measures on a set $\Omega$ Banach? With the total variation norm. I don't know what convergence means in this norm.. I mean how do I even think about it.
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352 views

Is there a non-reflexive Banach space which is strictly convex?

I just come up with the fact that a space being strictly convex, does not implies it is reflexive (at least I never saw a proof of it). How can one construct a example of a non-reflexive Banach ...
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1answer
318 views

Convergence of functionals and weak convergence

I consider a Banach space $V$ with its dual $V'$. I had a sequence of functionals $\{f_k\}_{k\in \mathbb N} \subset V'$, and I wanted to show (strong or norm) convergence of $f_k \to f \in V'$. I ...
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79 views

Image of the tensor product of strict maps of Banach spaces

Let $f:A\to C$ and $g:B\to D$ be bounded linear maps of Banach spaces with closed image. Will $f\widehat{\otimes}g:A\widehat{\otimes}_\pi B\to C\widehat{\otimes}_\pi D$ also have closed image? What ...
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192 views

Equivalent conditions for weak and weak-$*$ convergence

Say $E$ is a normed space over the field $\mathbb K$ ($\mathbb R$ or $\mathbb C$) and $E^{*}$ its dual space. The notations for weak and weak - $*$ convergence are $x_{n} \xrightarrow{w} x$ and ...
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135 views

The Banach space $c_0$ is $C^{\infty}$-smooth.

In this paper, J. Eells defines this notion of $C^r$-smoothness for Banach spaces: A Banach space $E$ is $C^r$-smooth, $r \geq 0$, if there exists a nontrivial (that is, nonzero) $C^r$ function ...
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Proving the $l_p$ space is complete.

I'm trying to prove $l_p$ spaces are complete. We have an $l_p$ space $W$. Let us take a cauchy sequence. There exists $N_0\in\Bbb{N}$ such that for $m,n>N_0$, $d(x^m,x^n)<\epsilon$. This ...
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prove that $(\mathbb{R}^n,\|\cdot\|_\infty)$ is complete.

Suppose I have the metric spaces $(\mathbb{R}^2,\|\cdot\|_2)$ and $(\mathbb{R}^n,\|\cdot\|_\infty)$ where $\|x-y\|_2=\sqrt{\sum_{i=1}^2 (x_i-y_i) }$ and $\|x-y\|_\infty =\max ...
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closed subspaces of locally convex inductive limits

Let $\left(V_{n},\phi_{n,n+1}\right)_{n\in\mathbb{N}}$ be an inductive sequence of LCTV spaces. A locally convex inductive limit of $\left(V_{n},\phi_{n,n+1}\right)_{n\in\mathbb{N}}$ is it's ...
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Reflexivity of $X \times Y$

I want to prove the following Theorem. Let $X,Y$ be reflexive. Then $X \times Y$ is reflexive. Here my try. Proof. Let $J_X, J_Y$ be the canonical injections of $X$ onto $X''$ and of $Y$ onto ...
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1answer
226 views

Is this proof correct? (left inverse and topologically complementary subsets)

I want to prove the following theorem: Theorem. Assume $T \in \mathcal L ( X, Y )$ is injective. The following statements are equivalent: $T$ admits a left inverse; Im($T$) is closed and ...
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Does one have to check some additional hypothesis to apply the implicit function theorem for infinite dimensional spaces?

Let $L : B_1 \rightarrow B_1$ be an isomorphism of Banach spaces (i.e., L is a bijective bounded linear operator) and $A: B_1 \rightarrow B_2$ a non linear operator. Consider the following equation ...
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1answer
120 views

Supremum of measurable function

Let $X$ be a Banach space and for each $t \in [a,b]$ let $Y_t$ be a Banach space. Let $F_t:X \to Y_t$ be a bounded map for each $t$. I know that for given $u \in X^*$ and for all $w \in X$, ...
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Bounded operators with prescribed range - part II

This is a continuation of the question bellow, in a more particular case. Bounded operators with prescribed range If $X$ is a separable Banach space and $Y$ is a closed, infinite dimensional ...
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Non strictly singular operators

Let $X$ be a separable Banach space and let $T:X\to X$ be a bounded operator that is not strictly singular. Can we always find an infinite dimensional, closed, and complemented subspace $Y$ of $X$ ...
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556 views

Proof of existence of Schauder basis for $L^p(\Omega)$?

There are a statements around, see [Brezis 2011, p. 146], like All classical (separable) Banach spaces used in analysis have a Schauder basis . I was wondering where to find a proof confirming ...
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166 views

Absolutely convergent but not convergent in not complete vector spaces

It is well known that in Banach spaces absolute convergence implies convergence. Since this is a characterization of Banach spaces, it does not hold in others kinds of spaces. In particular, let we ...
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Can one use the IFT on Banach spaces and the simple harmonic oscillator to say that there is a solution for the motion of a pendulum?

Let $W^{k,p} (S^1)$ be the Sobolev space of $k$ times weakly differentiable periodic functions, all whose weak derivatives upto $k$ are in $L^p(S^1)$. Consider the non linear operator $A_t : W^{k,p} ...
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Equivalent definitions of Injective Banach Spaces

A Banach space $X$ is said to be injective if for all Banach spaces $W,Z$ with $W\subset Z$, and operators $T\in B(W,X)$, $T$ can be extended to all of $Z$ with the same norm. Equivalently, $X$ is ...
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88 views

Extend a linear functional on “nice” functions to an integral

I have a positive linear functional $h$ defined on a set of Lesbesgue-measurable functions of "moderate growth" on $\mathbb{R}^2$–call this set $MG(\mathbb{R}^2)$. (A function $f$ is positive if ...
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Proof that the normed space $(l^{\infty},\|\cdot\|_{\infty})$ is Banach space

I have to prove that $(l^{\infty},\|\cdot\|_{\infty})$ is Banach space and I have some difficulties. This is what I've done. $l^\infty=\{x=\langle x_k\rangle, k\in N|\exists M>0 \ such\ ...
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388 views

Weak sequential completeness

It is obvious that reflexive spaces are weakly sequentially complete. Can we have a kind of a converse to this fact? Is there a non-reflexive Banach space $X$ such that both $X$ and $X^*$ are weakly ...
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Is the space $\ell^2_1$ injective? ($\ell^2_1$ = 2 dimensional(complex) space with 1-norm)

A Banach space $Z$ is said to be injective if for for any bounded linear map $\varphi: X \rightarrow Z$ and for any Banach space $Y$ containing $X$ as a closed subspace, there exists a bounded ...
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283 views

A distance-minimizing continuous projection onto a finite-dimensional subspace?

Let $E$ be a Banach space, which need not be a Hilbert space, and let $F$ be a finite-dimensional subspace of $E$. Suppose that for all $x \in E$, there exists a $y \in F$ realizing the minimal ...
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Defining multiplication and differentiation in a dual space

Let $B$ be a space of complex-valued analytic functions from some compact subset $C$ of $\mathbb{C}^n.$ $B$ is a Banach space with the uniform norm. Now, the paper I am reading says that we would ...
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92 views

Do I have a Banach space given the following norm?

This is very very similar to my other question asked three months ago. That time there was no Banach space because an integral in the norm definition allowed a counter-example. Once again I have a ...
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1answer
191 views

What is a decreasing scale of Banach spaces?

I am a having a hard time understanding a part of an article I am reading. The screen-cap is below. Basically, it's the line labeled (6) that I do not understand. I am not familiar with the circular ...
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Banach space integral via defining it in $X^{**}$ and then proving it's in $X$

Vector-valued integration is something I generally try not to think about very much. I have the impression that it can be a sort of "rabbit hole" of a subtlety if one allows it to be. So, I tend to ...
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Bounded operator and dense sets

Let $A : E \rightarrow E$ be a linear operator, and $E$ a (typically infinite-dimensional) Banach space. Suppose that $S$ is a collection of norm 1 elements of $E$ spanning a dense subspace of $E$. ...
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A basic question on the linear operators on Banach space

Given that $\cal{X}$ and $\cal{Y}$ are Banach spaces. $A:\cal{X} \to \cal{Y}$ and $Y:\cal{Y}\to \cal{X}$ are linear bounded operators. If $\rho(I_{\cal{X}} - YA)<1$, $\rho(I_{\cal{Y}} - ...
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Why if $\rho(I_{\mathfrak{X}} - YA)<1$ then $YA$ is invertible on the $R(YA)$?

I am reading an article where I am stucked at one point. Below is my problem. Given that $\mathfrak{X}$ and $\mathfrak{Y}$ are Banach spaces. $A:\mathfrak{X} \to \mathfrak{Y}$ and ...
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Distance of a point from a subspace vs. diameter

Let X = $(\Bbb R^N, \|\cdot\|)$ be a Banach space. Let $x_0 \in S^{N-1} = \{x \in \Bbb R^N : \sqrt{x_1^2+...+x_n^2}=1\}$. Denote $B^N_2 = \{x \in \Bbb R^N : \sqrt{x_1^2+...+x_N^2} \le 1 \}$. Define ...
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Need help with one basic result of linear operators on Banach spaces

$X$ and $Y$ are Banach spaces and suppose that $T$ be a closed subspace of $X$. $A:X \to Y$, $B:Y\to X$ and $X_{0}:Y\to X$ are linear bounded operators. It is given that the operator $BA$ is ...
3
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Uncountable basis and separability

We know that a Hilbert space is separable if and only if it has a countable orthonormal basis. What I want to ask is If a Hilbert space has an uncountable orthonormal basis, does it mean that it is ...
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1answer
239 views

Show that ($\ell^1$, $\|\cdot\|_1$) is complete

Show that the vector space $\ell^1 : = \{(a_n) : \sum_n|a_n| < \infty\}$ with the norm $\|(a_n)\|_1 : = \sum_n|a_n|$ where $(a_n)$ are sequences in $\mathbb C$ is complete. Thanks in advance.
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Distance from a point to a plane in normed spaces

How can we calculate the distance from a point to a plane in normed spaces ? where we are not inner product, for example: Calculate the distance in $(C[0,1],||\cdot||)$ endowed with the supremum norm ...