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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3answers
202 views

How to show convexity of a ball in metric space?

If $(X,\|\cdot\|)$ is a normed linear space, then how to show any ball $B(x,r)$ is convex? I know that if $x,y\in A\subset V$ then $[x,y]\subset A$, where $A$ is a convex subset of vector space $V$ ...
4
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4answers
328 views

A question about applying Arzelà-Ascoli

An example of an application of Arzelà-Ascoli is that we can use it to prove that the following operator is compact: $$ T: C(X) \to C(Y), f \mapsto \int_X f(x) k(x,y)dx$$ where $f \in C(X), k \in C(X ...
1
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1answer
139 views

Reproducing Kernel Hilbert Space- notation and basics

Am reading about Reproducing Kernel Hilbert Space(RKHS) while reading through Functional Analysis and Hilbert Space material and am unable to get the notation : $k(·,xi)$ correctly. What does the dot ...
5
votes
2answers
917 views

Subspaces of separable normed spaces

Let $X$ be a separable normed space. Is it true that every subspace is separable? If it was Hilbert space I would take the dense set and then their projections. It sounds trivial but I cannot prove ...
11
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2answers
587 views

How to show that $\mathbb R^n$ with the $1$-norm is not isometric to $\mathbb R^n$ with the infinity norm for $n>2$?

Could you please give me a hint to prove that $\mathbb{R}^n$ with the 1-norm $\lvert x\rvert_1=\lvert x_1\rvert+\cdots+\lvert x_n\rvert$ is not isometric to $\mathbb{R}^n$ with the infinity-norm ...
6
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1answer
217 views

If the unit sphere of a normed space is homogeneous is the space an inner product space?

Consider a normed vector space $V$. Suppose that for every pair of unit vectors $v,w$ there exists a linear isometry which sends $v$ to $w$ (and leaves the subspace spanned by $v$ and $w$ invariant). ...
2
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1answer
512 views

Question about proof that $C(X)$ is separable

In my notes we prove Stone-Weierstrass which tells us that if we have a subalgebra $A$ of $C(X)$ such that it separates points and contains the constants then its closure (w.r.t. $\|\cdot\|_\infty$) ...
3
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1answer
471 views

Linear isometry and operator norm $=1$

For some reason I used to think that if $T$ is a linear operator on normed spaces $V \to W$ then saying $T$ is an isometry is the same as saying $\|T\|_{op} = 1$. Well, I got stuck on a proof and ...
7
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2answers
642 views

Domain of an operator in functional analysis

I used to think that if we say $f$ is a function from a set $X$ to $Y$ then this implied that $f$ was defined on all of $X$. Because the definition of function is that it's a set $\{(x,y) \mid \text{ ...
5
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3answers
224 views

function from non separable normed space $X$ onto separable normed space $Y$.

Can you please give me a simple example of linear continuous mapping from non separable normed space $X$ onto separable normed space $Y$. Thanks a lot.
3
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1answer
138 views

Prove $\|F(x,·)\|_{L^{4,1}(E)}^2 \leq C\|F(x,·)\|_{L^{\infty}(E)}\|F(x,·)\|_{L^{2,2}(E)}$

How to derive this inequality? $$\|F(x,·)\|_{L^{4,1}(E)}^2 \leq C\|F(x,·)\|_{L^{\infty}(E)}\|F(x,·)\|_{L^{2,2}(E)},$$ where $C$ is constant and ...
1
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1answer
45 views

Gaussian type and Euclidean sections

I have a second question about Chapter 9 in Milman and Schechtman's book "Asymptotic theory of finite dimensional normed spaces" (first question here). It's about the proof of Theorem 9.7 (pg. 55). ...
1
vote
2answers
129 views

Does $(f,Tg)_{L^2}$ define an inner product space?

To my understanding inner product $$(f,g)_{L^2(\mathcal{D})} = \int_\mathcal{D} f(\boldsymbol{x})g(\boldsymbol{x})\,\mathrm{d}\boldsymbol{x},~~\mathcal{D} \subset \mathbb{R}^N$$ defines an inner ...
2
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3answers
157 views

Question about proof that multiplication in Banach algebra is continuous

Here's the proof in my notes: Where does the last inequality come from? If I want to show that it's continuous at $((x,y)$ I can use the inverse triangle inequality to get $$ (\|x^\prime\| + ...
3
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0answers
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 ...
0
votes
0answers
224 views

Unique continuous extension of $\tilde{L}: W \to Z$

Can you read my proof and tell me if it's correct? Thank you! Let $W$ be a dense subset of a normed vector space $V$ and let $\tilde{L}: W \to Z$ be a bounded linear operator into a Banach space. ...
3
votes
1answer
321 views

Computing operator norm exercise

I did the following exercise (given in my notes) can you tell me if my answer is correct? Thanks. Exercise: Compute the operator norm of the continuous map $f \mapsto f$ when viewed: (a) as a map ...
1
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3answers
128 views

$B(V,W)$ is complete if $W$ is

Let $B(V,W)$ be the space of bounded linear maps from $V$ to $W$. Then it is complete with respect to the operator norm. Can you tell me if my proof is correct? Thanks. It's easy to verify that the ...
1
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1answer
68 views

Euclidean sections of normed spaces with known cotype

I'm having trouble digesting the proof of Theorem 9.6 in Milman and Schechtman's classic book "Asymptotic theory of finite dimensional normed spaces" (pg. 54). I'm new to functional analysis, so this ...
0
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0answers
241 views

$C_c(X)$ dense in $L^p$

In class we proved that $C_c(X)$ is dense in $L^p$ where $X$ is a locally compact, $\sigma$-compact Hausdorff space either equipped with a Radon measure or equipped with a locally finite measure ...
4
votes
3answers
392 views

Simple application of Stone-Weierstrass

I was looking for a simple application of the Stone-Weierstrass theorem. First I thought that if $X$ is any compact measure space then Stone-Weierstrass implies that $C_c(X)$ is dense in $L^p$. But ...
2
votes
1answer
93 views

Balls and transformed sets in normed vector spaces

Let $T$ be a surjective, continuous linear operator between two Banach spaces $E$ and $F$. Assume that it is $B_F(y_0,4c)\subset \overline{T(B_E(0,1))}$, where $c>0$, $y_0 \in F$ ($B$ is for ...
2
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0answers
205 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? ...
3
votes
2answers
138 views

Proof of the lemma used in proving that a finite-dimensional normed space is complete

I'm trying to understand the proof for the lemma: $$\|\alpha _1 e_1 + \alpha _2 e_2 + \cdots + \alpha_n e_n\| \geq c (|\alpha_1|+|\alpha_2|+\cdots+|\alpha_n|)$$ where $c>0$ and the $e_i$s are ...
1
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1answer
256 views

Question about proof of Arzelà-Ascoli

(Arzelà-Ascoli, $\Longleftarrow$) Let $K$ be a compact metric space. Let $S \subset (C(K), \|\cdot\|_\infty)$ be closed, bounded and equicontinuous. Then $S$ is compact, that is, for a sequence $f_n$ ...
3
votes
1answer
140 views

The trace norm cannot be increased by composing with a unitary operator

$\newcommand{\tr}{\operatorname{tr}}$ I was reading a proof for the statement $|\tr(US)|\leq |\tr(S)|$, for every endomorphism $S$ on a complex vector space $H$ and every unitary operator $U$ on the ...
2
votes
1answer
319 views

Completion of $C_c$ with respect to $\|\cdot\|_\Psi$

I'm doing the second half of the following exercise in my lecture notes: "Let $C_c(R)$ be the vector space of continuous functions $f : R \to R$ with $\mathrm{supp}(f)=\overline{ \{x \in R \mid ...
2
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1answer
208 views

Completion of $C_c(X)$ with respect to $\|\cdot\|_\infty$

Notation: All functions here are from $X$ to $\mathbb R$. $C_c(X)$ = compactly supported continuous functions. $C_b(X)$ = bounded continuous functions. $B(X)$ = bounded functions. $C_0(X)$ = ...
9
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2answers
213 views

If $V \times W$ with the product norm is complete, must $V$ and $W$ be complete?

Let $V,W$ be two normed vector spaces (over a field $K$). Then their product $V \times W$ with the norm $\|(x,y)\| = \|x\|_V + \|y\|_W$ is a normed space. Using this norm it's easy to show that if ...
1
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1answer
191 views

$C_0(X)$ is a closed subspace of $C_b(X)$

Can you tell me if my proof is correct? Thank you! Claim: $C_0(X)$ is a closed subspace of $C_b(X)$ Proof: We have to show that $C_0(X)$ contains all of its limit points. Let $f(x)$ be a limit point ...
1
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1answer
107 views

Typo in lecture notes?

The following is an example in my lecture notes: "Let $X$ be a locally compact topological space (that is, a topological space in which every point has a compact neighborhood). Then $C_0(X)=\{f \in ...
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2answers
159 views

Seminorm exercise

Can you tell me if my answer is correct? It's another exercise suggested in my lecture notes. Exercise: Consider $C[-1,1]$ with the sup norm $\|\cdot\|_\infty$. Let $$ W = \{f \in C[-1,1] \mid ...
1
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1answer
378 views

Examples of seminorms

Let $V$ be a vector space with a seminorm $\|\cdot\|_s$. Then apparently we can turn $\|\cdot\|_s$ into a norm $\|\cdot\|$ on $V/W$ by defining $\|v + W\| = \inf_{w \in W} \|v + w\|_s$ where $W$ is ...
5
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0answers
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
votes
4answers
367 views

Product norm on infinite product space

Today I proved that if $V$ is a normed space with norm $\|\cdot\|$ then I can define a norm on $V \times V$ that induces the same topology as the product topology as follows: $\| (v,w) \|_{V \times V} ...
2
votes
1answer
707 views

Proof that every normed vector space is a topological vector space

The topology induced by the norm of a normed vector space is such that the space is a topological vector space. Can you tell me if my proof is correct? Of course we have to show that addition and ...
1
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1answer
67 views

Something weaker than the Riesz basis

I have some function $f$, real valued and continuous. I formed functions $\{f_{m,k}, k \in \mathbb{Z}, m>0\}$ such that that $\mathrm{span}\{f_{m,k}, k \in \mathbb{Z}, m>0\}$ is dense in ...
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1answer
116 views

Practical implications of a vector space being a topological vector space

I have a space $V$ and I lately discovered that it's a topological vector space. What are the practical implications of that?
1
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1answer
42 views

How to show growth without bound in only certain cases and not in others?

I encountered the following problem. (We're working in a finite-dimensional real vector space, here.) Suppose $$A=\frac{1}{2}\left(\begin{array}{cc}-2 & 4\\1 & 1\end{array}\right).$$ Find ...
2
votes
3answers
272 views

Cauchy in Norm and Weakly converge Implies Norm convergent

Let $X$ be a normed space and $(x_n)$ is a Cauchy sequence in the norm sense. Also assume the $x_n \rightarrow x_0 $ weakly. Then $x_n \rightarrow x_0 $ in norm. What I did:Take $ \varepsilon ...
2
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1answer
139 views

Banach space, Normed vector space

Help me please with this question. Let's $Y$ be Banach space, $Z$ - Normed vector space and $(T_{n})_{\mathbb{N}}$ - the sequence in $B(Y,Z)$ so that all sequence $(y_{n})_{\mathbb{N}}$ in Y holds: ...
6
votes
1answer
1k views

An example of a norm which can't be generated by an inner product

I realize that every inner product defined on a vector space can give rise to a norm on that space. The converse, apparently is not true. I'd like an example of a norm which no inner product can ...
2
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1answer
82 views

Norms on inner product space over $\mathbb{R}$

Definition of the problem Let $\left(E,\left\langle \cdot,\cdot\right\rangle \right)$ be an inner product space over $\mathbb{R}$. Prove that for all $x,y\in E$ we have $$ \left(\left\Vert ...
2
votes
1answer
506 views

Cauchy-Schwarz Inequality proof (for semi-inner-product A-module).

I am reading a proof of the following Cauchy-Schwarz Inequality and I don't understand one part of the proof: Theorem: Let $A$ be a $C^*$-algebra and let $E$ be a semi-inner-product $A$-module. Then ...
8
votes
1answer
111 views

Inner product space over $\mathbb{R}$

Definition of the problem I have to prove the following statement: Let $\left(E,\left\langle \cdot,\cdot\right\rangle \right)$ be an inner product space over $\mathbb{R}$. prove that for all $x,y\in ...
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0answers
69 views

Is there chance to form a frame (Riesz basis)?

Let $f$ be a continuous function on $\mathbb{R}$ with compact support with exactly one maximum. Form the functions $$ f_{m,k}(x)=f^m\left(x-\frac{k}{2^m}\right) $$ One can show that ...
2
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1answer
276 views

continuous projections to finite dimensional subspaces of normed spaces

If $X$ is a normed space and $Y$ is a finite dimensional subspace, then there exists a continuous linear projection $P$ from $X$ to $Y$. Our teacher gave us the instruction to use the following fact: ...
5
votes
2answers
358 views

If $ x_n \to x $ then $ z_n = \frac{x_1 + \dots +x_n}{n} \to x $

I have this problem I'm working on. Hints are much appreciated (I don't want complete proof): In a normed vector space, if $ x_n \longrightarrow x $ then $ z_n = \frac{x_1 + \dots +x_n}{n} ...
3
votes
1answer
92 views

What are the $n$-th degree minimal polynomials for $L^p([-1,1])$?

It is known (even by me) that the Chebyshev polynomial of degree $n$ (of the first kind) is the minimal polynomial in the space $L^{\infty}([-1,1])$ for a fixed $n$ and leading coefficient $2^n$. ...
1
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0answers
69 views

Dense property of $C^k_0(\Omega)$

When $\Omega = \mathbb{R}^d$ or $\mathbb{R}^d_+$, $C^k_0(\bar{\Omega})$ is dense in $W^{k,p}(\Omega)$(Sobolev space, k-differentiable and each term estimated by p-norm). I am wondering if it holds ...