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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2
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2answers
571 views

Series in incomplete normed space

We have known that "A normed space $X$ is a Banach space if and only if each absolutely convergent series in X converges". We would like to find an explicitly incomplete normed space and an explicitly ...
1
vote
2answers
123 views

My “wrong” comparison between $\ell^2$ and $\ell^1$

For sure, $\ell^2$ is larger than $\ell^1$, because for $|x|<1$, $|x|^2<|x|,$ that is, $||x||_2\leq||x||_1.$ But using Cauchy-Schwartz inequality, I get a "wrong" comparison: ...
2
votes
2answers
121 views

on proving that $\|\cdot\|_2$ is a norm on $C[0,1]$

Let $\mathbb{F}$ be either $\mathbb{R}$ or $\mathbb{C}$ and consider the vector space $C[0,1]$, the collection of continuous functions $f\colon[0,1]\to\mathbb{F}$. I want to show that $\|\cdot\|_2$ is ...
1
vote
1answer
121 views

convergence in function space

Maybe is a silly question, but for some reason I am confused... If $\mathcal{F}$ is a normed space of real functions and $\displaystyle{ f \in \mathcal{\bar F } }$ then there exists a sequence of ...
0
votes
1answer
43 views

integrable, $L_1$ and $L_\infty$

I have a question about normed space and integrable. If $f$ is in $L_\infty$, $g$, which is $g \le f$, can be absolutely integrable ($g$ is in $L_1$)? And how can I prove it?
4
votes
1answer
121 views

Convergence in $L_\infty$ and $L_1$ even if infinite measure space

Let $\{f_n\}_{n\in \mathbb{N}}$ be a sequence of measurable functions on a measure space and $f$ measurable. In the literature, assuming the measure space $X$ has finite measure, if $f_n$ converges to ...
1
vote
2answers
167 views

Intersection of a unit sphere of a given norm in finite dimension with an hyperplane.

Let $\|\cdot\|$ be a norm on $\mathbb{R}^n$. Let $C:=\{x\in\mathbb{R}^n\,:\,\|x\| \leq 1\}$, that is to say let $C$ be a convex compact symmetric set of non empty interior. Let $H$ be a linear ...
1
vote
5answers
186 views

Is closure of convex subset of $X$ is again a convex subset of $X$?

Today I was going through my text book exercise and got hold of the following question. Let $X$ be a normed linear space with norm $\lVert\cdot\rVert$ and $A$ is a non empty convex subset of $X$ ...
2
votes
2answers
876 views

Is closure of linear subspace of X is again a linear subspace of X??

Let $X$ be a normed linear space with norm $||\cdot||$ and $A \neq \emptyset$ is a linear subspace of $X$. Prove that $\bar{A}$ is also a linear subspace of $X$.
1
vote
2answers
435 views

Show that the discrete metric can not be obtained from $X\neq\{0\}$

If $X \neq \{ 0\}$ is a vector space. How does one go about showing that the discrete metric on $X$ cannot be obtained from any norm on $X$? I know this is because $0$ does not lie in $X$, but I am ...
1
vote
1answer
350 views

Finding a cauchy sequence that does not converge on M

We define the following infinity norm on $\mathbb{R}$ as follows $$l_\infty(\mathbb{R}) = \{ (x_i)_{i \in \mathbb{N}} \,\mid\, x_i \in \mathbb{R}, \sup_{i\in\mathbb{N}} \left|x_i\right|<\infty \}$$ ...
1
vote
1answer
38 views

Replacing one of the conditions of a norm

Consider the definition of a norm on a real vector space X. I want to show that replacing the condition $\|x\| = 0 \Leftrightarrow x = 0\quad$ with $\quad\|x\| = 0 \Rightarrow x = 0$ does not alter ...
3
votes
1answer
119 views

Question about norms and coarseness of topology

I've been thinking about norms and asked myself the following question: If I have two norms $\|\cdot\|_A$ and $\|\cdot\|_B$ with $\|\cdot\|_A \leq \|\cdot\|_B$, which topology is coarser, that is, ...
5
votes
1answer
622 views

Equivalence of reflexive and weakly compact

In a normed space $X$ is there an equivalence between these two proposition? $1)$ $X$ is reflexive; $2)$ $B$, the unit ball of $X$, is weakly compact.
49
votes
2answers
1k views

Explicit norm on $\mathcal{C}^0(\mathbb{R},\mathbb{R})$

Do you know an explicit norm on $\mathcal{C}^0(\mathbb{R},\mathbb{R})$? Using the axiom of choice, every vector space admits a norm but have you an explicit formula on ...
1
vote
1answer
348 views

Distance between real finite dimensional linear subspaces

Is there a usual distance between linear subspaces ($V,W$) of an n-dimensional normed vector space with inner product? In the case of hyper-planes one could use the angle (based on the inner product ...
2
votes
1answer
240 views

$C^1 [0,1]$ with different norm

If the space $C^1 [0,1]$ is equiped with norm $\Vert \cdot\Vert_1$,where $$ \Vert f\Vert_1=|f(0)|+\Vert f'\Vert _{C}=|f(0)|+\sup_{t\in [0,1]}|f'(t)| $$ for any $f\in C^1 [0,1]$, is this space Banach? ...
1
vote
0answers
208 views

equivalence of norms in an open set

I would like to prove that two given norms in the space of smooth functions are equivalent in an open set, is it enough to show that they are equivalent for any compactly contained open set? why? ...
2
votes
1answer
218 views

inequality between norms of vector

Let $a=(a_1, \ldots, a_n)$ be a vector in $R^n$. It is known that $\|a\|_{\infty}\leq \|a\|_2\leq \sqrt n\|a\|_{\infty}$. Let $k<<n$. For which kind of vectors the following would be true: $$ ...
2
votes
3answers
184 views

If $(x_n)$ converges weakly to $x$, then $x$ is in the closure of the span of the $x_n$

I need your help with this problem that I founded it in a lecture notes. Then, the problem says: Let $ X $ be a normed space. Show that if a sequence $ (x_n) _ {n \in \mathbb {N}} $ in $ X $ ...
1
vote
3answers
179 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
votes
4answers
302 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
vote
1answer
125 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
796 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
votes
2answers
518 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
votes
1answer
205 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
votes
1answer
453 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
votes
1answer
409 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
votes
2answers
522 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
votes
3answers
214 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.
0
votes
0answers
150 views

When the linear operator is continuous.

Could I have a hint please on how to prove the following proposition: Let $X$ and $Y$ be two normed space and $T$ be a linear operator from $X$ into $Y$. The operator $T$ is continuous if the ...
3
votes
1answer
136 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
vote
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
126 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
votes
3answers
136 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
votes
0answers
114 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
215 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
278 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
vote
3answers
125 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
vote
1answer
67 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
votes
0answers
214 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
361 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
90 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
votes
0answers
198 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
132 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
vote
1answer
248 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
130 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
294 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
votes
1answer
194 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
votes
2answers
206 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 ...