A vector space $E$, generally over the field $\mathbb R$ or $\mathbb C$ with a map $\lVert \cdot\rVert\colon E\to \mathbb R_+$ satisfying some conditions.

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114 views

Operator norm of a convolution

Consider the operator on $L^2(\Bbb R)$, $f\rightarrow f*g$, where $g\geq 0$ is some $L^1$ function. Show the operator is a bounded linear operator with operator norm equal to $||g||_1$. Showing ...
5
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302 views

Are Arzelà–Ascoli theorems results of similar theorems on normed spaces, metric spaces or other spaces?

From Wikipedia, two generalizations of the Arzelà–Ascoli theorem are Let $X$ be a compact Hausdorff space. Then a subset $F$ of $C(X)$, the set of real-valued continuous functions on X, is ...
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71 views

Find Lipshitz constant

Let $x, y$ be two vectors in $R^n$. Let $\pi$ be permutation on $\{1, \ldots, n\}$ with uniform distribution. Find Lipshitz constaants of the following functions: $$ f(\pi)=\sum_{i=1}^nx_{\pi(i)}y_i ...
4
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106 views

What are norms used for?

These two questions are quite similar to this one, so I apologise if this irritates anyone. Also, I suspect that a lot can be said in the answer, so I am really just looking for some main points ...
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51 views

Condition on Equality of closure of open ball and closed ball, suppose I have a counterexample

Let $(X, d)$ be a metric space. Also for $x \in X$ and $r \ge 0$ define: $$ B(x,r) = \{ y \in X : d(x,y) < r \} \quad \mbox{ and } \quad K(x,r) = \{ y \in X : d(x,y) \le r \}. $$ Denote by ...
3
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27 views

give necessary and sufficient conditions that every functional in $w^*-cl M$ be the $w^*$- limit of a sequence from M

Let $X$ be a separable Banach space. If $M$ is a linear manifold in $X^*$ give necessary and sufficient conditions that every functional in $w^*-\mathrm{cl} M$ be the $w^*$- limit of a sequence from ...
3
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82 views

How to prove that this set is closed.

Lets say $a_1, a_n$ are normed vectors. Why is the set $C = \{\Sigma_{i=1}^n \lambda_ia_i: \lambda_i \ge0\}$ closed? The $\lambda$'s can be any non-negative numbers. So C is the set of all ...
3
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107 views

Asymptotically isometric copy of $\ell_1$

A Banach space $X$ is said to contain Asymptotically isometric copy of $\ell_1$ if there is a null sequence$(\rho_n)_{n=1}^{\infty}$ in $(0,1)$ and a sequence$(x_n)_{n=1}^{\infty}$ in $X$ so that ...
3
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0answers
48 views

What extra assumption makes this transformation affine?

Let a vector space $V$ be given. Let $f:V\to V$ have the property that for all $x,y,a\in V$, $$ f(x+a)-f(y+a) = f(x) - f(y) \tag{$\star$} $$ Q1. I'd like to know how weak one can make additional ...
3
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62 views

why is test function space $\mathcal{A}$ complete

I am trying to find out, why the space $$\mathcal{A}:=\left\{\phi\in C_0(\mathbb{R}^{2d})|\;\|\phi\|_\mathcal{A}:=\int_{\mathbb{R}^d}\sup_{x\in\mathbb{R}^d}|(\mathcal{F}_p\phi)(x,y)|\;\mathrm ...
3
votes
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134 views

Two norms $\|\cdot\|_a$ and $\|\cdot\|_b$ on $X$, and a function $f:X\to Y$ Fréchet differentiable with one of the norms but not with the other one?

There is a theorem that if $f : (X,\|\cdot\|_{X1}) \to (Y,\|\cdot\|_{Y1}) $ is Fréchet differentiable, then replacing the norms with some equivalent norms $\|\cdot\|_{X2}$ and $\|\cdot\|_{Y2}$ ...
3
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91 views

Is $\rho(f) := \mu(|f| \geq \epsilon), \epsilon >0$ some kind of generalized norm?

Let $\rho(f) := \mu(|f| \geq \epsilon)$ where $\epsilon >0$ is a constant and $\mu$ is a measure and $f$ is a measurable function. Then $f\equiv 0 \rightarrow \rho(f) = 0$ when $|a| \neq 0$, ...
3
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115 views

Do the $p$-norms in $\mathbb R^2$ have a nice geometric intuition behind them?

For $p\geq 1,$ the $p$-norm of a vector $(x,y)\in\Bbb R^2$ is the number $\|(x,y)\|_p=(|x|^p+|y|^p)^{1/p}.$ I learned this definition some time ago, but I never really understood it. Is there a ...
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2k views

Proof that every finite dimensional normed vector space is complete

Can you read my proof and tell me if it's correct? Thanks. Let $V$ be a vector space over a complete topological field say $\mathbb R$ (or $\mathbb C$) with $\dim(V) = n$, base $e_i$ and norm ...
2
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26 views

Extension of function with values in a Banach space

I want to prove the following Let $E,X$ be Banach spaces, and $Y\subset E$ a closed subspace with codimension $1$. Let $T:Y \to X$ be a continuous linear function. Then there exists a continuous ...
2
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33 views

Adjoint of an operator on $C(X)$

Let $X,Y$ be compact and $\tau: Y\to X$ be continuous. If $A:C(X)\to C(Y)$ such that $(Af)(y)=f(\tau(y))$ is a linear and continuous operator, then show that the adjoint operator $A^*:M(Y)\to M(X)$ is ...
2
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31 views

Weakly closed convex set and norm convergent

Let $X$ be a normed space and let $\{x_n\}$ be a sequence in $X$ such that $x_n\to x$ weakly. Show that there is a sequence $\{y_n\}$ such that $y_n\in \operatorname{co}\{x_1,...,x_n\}$ and ...
2
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0answers
27 views

extension theorems on normed spaces

I know that there are a number of extension theorems, Tietze's extension theorem, Hahn-Banach extension and so on.. I want to know if there is an extension theorem which guarantees that if say $X$ is ...
2
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45 views

relation between of uniformly rotund in every direction and uniformly rotund and locally uniformly rotund

The norm of a Banach space $X$ is said to be uniformly rotund in every direction if $$\lim_{n→∞}\|x_n−y_n\|=0$$ whenever $$x_n,y_n∈SX$$ are such that $$\lim_{n→∞}\|x_n+y_n\|=2$$ and there is a $z∈X$ ...
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119 views

Convex subsets, Normed spaces, Separating hyperplane

I'm trying to understand the proof of a theorem taken from a textbook. Theorem: If $S_1,S_2$ are disjoint non-empty convex subsets of a real vector space $X$ (which may be infinite dimensional) then ...
2
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31 views

Normalizing multiple different features from unknown distributions

I'm doing some "exploratory" data analysis over a large set of classes/proteins, with a few hundred different features (I.E. Continuous variables) extracted from the data. The features are calculated ...
2
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63 views

Space of Continuous mappings to metric spaces

I want to ask whether some basic result from the space $C([0,1],R)$, where $R$ is the real space carries over to the space $C([0,1],E)$, where $(E,\|\cdot\|_E)$ is a metric space. We know that ...
2
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82 views

Linear Operators: Continuous $\Rightarrow$ Bounded

Let $T:V\rightarrow V'$ be a continuous linear operator between two normed vector spaces $V,V'$. Show that it is bounded. Continuity is defined as $\lim_{n}\|x_n-x\|=0\Rightarrow ...
2
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208 views

Prove: Completion of a normed space is a Banach space.

Me again with another (most likely) easy problem that is too hard for me as of now. Suppose we have a normed space $(X,||\cdot||)$ and a map $i:X\rightarrow \hat X$ with $\overline{i(X)} = \hat ...
2
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0answers
197 views

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 ...
2
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0answers
65 views

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 ...
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98 views

Which are nontrivial examples of analytical functions on Frechet spaces?

Let $X$ be a real linear topological space, which is a (separable) Frechet space, such that the topology on $X$ is generated by the countable family $\{p_n:n\in\omega\}$ of norms . A real-valued ...
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76 views

Are these two definitions for dual norm equivalent

Suppose there is a p norm, also is a function $\left\|x\right\|_p$, the dual function at $z$ of it is, $$\sup\limits_x\left\{ z^{T} x-\left\|x\right\|_p\right\}$$ The second is, $$ \sup\limits_x ...
2
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0answers
43 views

Finding an orthornormal basis given a bilinear form

Take $ \langle \cdot,\cdot\rangle$ to be a nondegenerate symmetric bilinear form (which may not be positive definite). Take $\Vert x\Vert = \vert \langle x,x\rangle \vert^{1/2}$. I want to show that ...
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47 views

Do these inequalities imply these inequalities? (Norms and squaring)

Suppose we have the inequalities involving norms $$\lVert f \rVert_{X}^2 \leq C_1(\lVert f \rVert_{Y}^2 + \lVert f \rVert_{Z}^2)$$ and $$\lVert f \rVert_{X}^2 \geq C_2(\lVert f \rVert_{Y}^2 + \lVert f ...
2
votes
0answers
116 views

$\ell^0$ and $\ell^{\infty}$ norms

Let $x \in S^{n-1}$ and such that its coordinates $|x_1|\geq \cdots \geq |x_n|$. Under which condition on $\|x\|_0$ the following inequality is true that $$\|x\|_{\infty}\leq \frac{1}{\sqrt ...
2
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0answers
184 views

Continuous, Bounded Normed Linear Spaces

For $f$ in $C[a, b]$, define $||f||_1 = \int_a^b |f|$. Show that this is a norm on $C[a, b]$. Also, show that there is no number $c\geq0$ for which $||f||_{\max}\leq c||f||_1$ for all $f$ in ...
2
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0answers
157 views

Is it a Banach space? If so what is its dual?

Let $(E_n)$ be a sequence of Banach spaces and $(w_n)$ be a sequence of positive real numbers. For $1\leq p <\infty$ define $\bigoplus\limits_P E_n:=\{(x_n):x_n\in E_n,\sum\limits_n\lVert ...
2
votes
0answers
199 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? ...
2
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0answers
90 views

Consequence of metrizability proof - disregard, the question is an error

In Marian Fabian et al's Functional Analysis and Infinite-Dimensional Geometry, Proposition 3.22 states/proves that if $X$ is a separable Banach space, then the (closed) unit ball, $B_{X^{*}}$ of ...
2
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0answers
125 views

norms on a vector space - is there a quicker way to approach this problem?

I'm looking at the following problem, which I have done but wonder whether there isn't a faster way of doing it. (It's a past exam question which is supposed to take 7.5 minutes, but I only managed to ...
2
votes
0answers
74 views

Embedding tree metric isometrically into $\ell_\infty$

I just started (independent) learning on metric embeddings from the Fall 2003 offering of the course at CMU. I have a limited mathematical background and alas, it made me stumble at the first exercise ...
2
votes
0answers
134 views

Different topologies on a normed/inner product space

Given a normed vector space $X$. Let $\mathcal{T}$ represent the topology on $X$ induced by the norm. Define: $A$:={topologies that can make the norm continuous}, $B$:={topologies that can make the ...
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0answers
257 views

Proof of equivalence in strictly convex spaces

I'm trying to understand a proof and from my current understanding there is one thing not clear/wrong. Maybe you can help: Let $(X, \lVert\cdot\rVert)$ a normed space and $B(0,1)$ the closed unit ...
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0answers
37 views

Proof of separability of Lp spaces

The following is a proof in Brezis book. It shows the separability of $L^{p}$ spaces: I have a few questions regarding the proof. Questions: It says 'it is easy to construct a function $f_{2} ...
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0answers
33 views

Cartesian product of reflexive spaces

Given $(E,\|\|_E),(F,\|\|_F)$ reflexive normed vector spaces. I have to prove that also $(E\times F,\|\|_{E\times F})$ is reflexive where $\|\|_{E\times F}$ is the product norm. What I know is that ...
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0answers
18 views

my question is about proving a norm is a norm

I want to show that $||f||=\max_{t\in [0,1]}|f(t)|$ defines a norm on $C([0,1])$. I only have a question on the triangle inequality property. This is what I have done but I am not quite sure it is ...
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0answers
22 views

Simultaneous extension and complemented subspace

The following is Exercise 3.13.5 of Conway's Functional Analysis: Let $X$ be a compact set and let $Y$ be a closed subset of $X$. A simultaneous extension for $Y$ is a bounded linear map $T:C(Y)\to ...
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0answers
54 views

Weak sequential compactness in a reflexive space

Let $\{X, \| \cdot \|\}$ be a normed space, $B$ is the unit ball of $X$. If $\{X, \| \cdot \|\}$ is reflexive, then is $B$ weakly sequentially compact? If it's not true, are there any counterexamples ...
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0answers
18 views

Terminology: Contraction *of* normed spaces? *Between* normed spaces? *On* normed spaces?

I have a terminological question. Suppose $X$ and $Y$ are normed spaces, and let $f$ be a contraction $X \to Y$. Which of the following expressions is correct? $f$ is a contraction of normed spaces. ...
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39 views

Finding the norm of a linear operation.

I am reading A course in real analysis by John McDonald, on page 530, it says "it is easy to show $|||J|||=1$" where $J$ is the linear operation $J:C([0,1])\rightarrow C([0,1])$, defined by $J(f)(x) = ...
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0answers
32 views

Does this topology on the dual have a name

Let $X$ be a topological vector space. Let $X^\ast$ denote its continuous dual. It is possible to endow $X^\ast$ with the weak star topology: Def.1: If $e_x: X^\ast \to \mathbb C$ is the map $\varphi ...
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0answers
26 views

A System of Matrix Equations

I am looking for the solutions to the following system of matrix equations: (1) $AA^{*} - A^{*}A = C^{*}C - BB^{*}$ (2) $DD^{*} - D^{*}D = B^{*}B - CC^{*}$ (3) $AC^{*} - C^{*}D = A^{*}B - ...
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vote
0answers
67 views

The Hilbert space $\mathcal{H}_\eta$ and unitary correspondence with $L^2[a,b]$

The question I have is related to a problem in Stein and Shakarchi's Real Analysis, Chapter 4. The problem Let $\eta(t)$ be a fixed strictly positive continuous function $[a,b]$. Define ...
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0answers
37 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 ...