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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5
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127 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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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 ...
5
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360 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 ...
4
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53 views

A normed space is not separable if and only if it contains an uncountable set of disjoint balls of the same radius

I want to show normed space $E$ is not separable iff $E$ contains uncountable set of pairwise non-intersecting balls of radius $r > 0$. Use contraposive: first prove $E$ is separable then ...
4
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78 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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123 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 ...
3
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0answers
47 views

Finding L^1 centers of sets of probability distributions

Let $\mathcal{P}^n = \{ x \in \mathbb{R}^n : x \geq 0, \sum x = 1\}$. Suppose I have $p_1, \ldots, p_m \in \mathcal{P}^n$. I want to find an $L^1$ center for these points. i.e. $q \in \mathcal{P}^n$ ...
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85 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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0answers
114 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
52 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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72 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
0answers
142 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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93 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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118 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 ...
3
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94 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
votes
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32 views

n- dimensional normed linear space isomorphic to n-dimensional Euclidean space

If $(X,\|\cdot\|)$ is an n- dimensional normed linear space over R. Is it isomorphic to n-dimensional Euclidean space $R^n$. I know it is topologically isomorphic but what about isometry? I think if ...
2
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0answers
13 views

Explicit Hahn-Banach extension formula in finite dimensional $l^p$ /Smoothness of the Hahn-Banach mapping

Consider the finite dimensional vector space $V=(\mathbb{R}^N,\|\cdot\|_{p})$, equipped with the usual $l^p$ norm, $1<p<\infty$. Consider a linear subspace $U\subset V$ (not necessarily a ...
2
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37 views

Gelfand width of $l_1$ unit ball in $\infty$ metric

The $l_1$ balls in $l^N_p$ are well studied, but I cannot find anything about estimates of $d^m(B^N_1,l^N_\infty)$ except of $d^m(B^N_1,l^N_\infty)\geq C\min\{1,\frac{\ln(eN/m)}{m}\}$ which is not ...
2
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44 views

Norm for a set of vectors

Let V be a normed vector space (real or complex valued) with norm $\|\cdot\|_V$. For any nonempty and bounded subset $A \subseteq V$ one can define $\|A\|$ via $$\|A\|:=\sup\{|x|:x\in A\}$$ I ...
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38 views

Proofchecking: Application of Banach-Alaoglu on weak converging nullsequence

Problem Assume $x_n \to 0$ weakly in a Banach space. Show that for all $\epsilon>0$ and for all $N\in \mathbb{N}$ there exists a $n>N$ s.t. for all $f\in X^\ast, \|f\|\leq 1$ there exists ...
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29 views

Metric space with measure and a special property

Let $R$ be a metric space endowed with a (complete) measure $\mu$ satisfying the following condition: all the open and closed sets of $R$ are measurable and for any measurable set $M\subset R$ and any ...
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votes
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22 views

Separable $X^*$ Property

Let $X$ be a normed space. If $X^*$ is separable, then there exists $(f_n)_{n\geq1}\subset X^*$ such that $\|f_n\|_{X^*}=1$ for all $n$ and $\{f_n:n\geq1\}$ is dense in $\{f\in X^*:\|f\|=1\}$. In ...
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135 views

Completely metrizable space, complete norm

Let $E$ be a normed, completely metrizable space. Prove that the initial norm is complete. How can I go about solving this problem? I will be grateful for all your hints. Thank you!
2
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35 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
votes
0answers
33 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 ...
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60 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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39 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
votes
0answers
48 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 $z∈X$ ...
2
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0answers
98 views

Complete Normed Space => Uncountable Hamel basis not by Baire

I need to show that a complete normed space X has no countable Hamel basis. One possibility is to with Baire's theorem. I, however, try to give an explicit sequence, namely: For a contradition, let ...
2
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0answers
129 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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0answers
39 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
votes
0answers
558 views

Natural matrix norm of an inverse matrix

Let $\left\|\cdot\right\| : \text{GL}(n,\mathbb{R})\to\mathbb{R}_{\ge 0}$ denote the natural matrix norm, i.e. $$\left\|A\right\|:=\max_{x\ne ...
2
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72 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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0answers
95 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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0answers
314 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
votes
0answers
288 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
votes
0answers
68 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 ...
2
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61 views

Sequence in a normed space exercise

Let $(a_n)$ be an arbitrary sequence in a normed space $E$. Show that there exists a sequence $(x_n)$ of points of $E$ such that $\,\,\,\displaystyle{\lim_{n\rightarrow \infty}x_n=0}\,\,\,$ and a ...
2
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0answers
109 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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0answers
85 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 ...
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0answers
45 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 ...
2
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0answers
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
63 views

Normed space Analysis

Show that the normed space $(\ell^2, \|\cdot\|_2)$ is complete. I am not sure where to even start with this question. I'm quite sure it involves Cauchy and also Bolzano-Weierstrass but I'm not sure ...
2
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122 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
198 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
votes
0answers
176 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
207 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
votes
0answers
131 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
79 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
151 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 ...