3
votes
0answers
34 views

Trouble finishing a (direct) proof that $\ell^2(A)$ is a complete metric space

Let $A$ be any non-empty set. We can define summations of non-negative numbers over this index set by using a supremum of summations over finite subsets of $A$. That is, $$\sum\limits_{\alpha \in A} ...
-1
votes
1answer
43 views

norm of integral operator in $C([0,1])$

If we define on $C([0,1])$ the operator $$ Tf(x) = \int_{0}^{1} K(t,s) f(s) ds$$ where $K$ is a continous function on two variables. I want to show that: $1)$ $||T|| = \displaystyle\max_{t} ...
1
vote
0answers
32 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 ...
1
vote
1answer
38 views

$X$ complete normed space $\implies\mathrm B(X,Y)$ complete normed space?

$\newcommand{\N}{\mathbf N}\renewcommand{\leq}{\leqslant}\renewcommand{\geq}{\geqslant} \newcommand{\eps}{\varepsilon}$I was looking through the functional analysis notes of TWK (on his webpage ...
0
votes
1answer
30 views

The dual of subspace of a normed space is a quotient of dual: $X' / U^\perp \cong U'$

I wanted to show that $X' / U^\perp \cong U'$, for $U$ being a closed subspace of the Banach space $X$. Therefore I looked at $l: X' / U^\perp \cong U' , x' + U^\perp=[x'] \mapsto x'|_U$. It is ...
0
votes
0answers
80 views

Strongly convex and Frechet differentiable function in reflexive Banach space

We first recall two definitions about strong convexity and Frechet differentiability in normed space. Let $(X, \|.\|)$ be a normed space and $f:X\rightarrow\mathbb{R}$ be a function. (a) $f$ is said ...
1
vote
1answer
33 views

Which of the following sets are open (or closed)?

a.) $A:= \{(x_n)_{n\in \mathbb{N}} : x_n \in [0,1] \hspace{2mm}\text{for all}\hspace{2mm} n\in\mathbb{N}\}$ in $(l^\infty, \|\cdot\|_{\infty})$ and b.) $B:= \{f\in C([0,1]) : |f(t)-t|<1 ...
0
votes
1answer
69 views

Separability of a normed space

The following is an Exercise 12, page 75 of conway's Functional Analysis. Let $\oplus_\infty X_i = \{x\in \sqcap X_i: ||x||=\sup||x(i)||<\infty\} $ where each $X_i$ is a normed space for $i\in I$. ...
2
votes
2answers
55 views

What's wrong in this reasoning of $l_\infty$ separability?

While solving a problem, related to functional analysis, I've accidentally got a "proof" of $l_\infty$ being separable, tried to find a fault in it (as the result isn't true), but didn't succeed in ...
2
votes
1answer
21 views

Continuous functional that separate points

This is an exercise from Royden's Real Analysis. Let $X$ be a normed linear space and $W$ a subspace of $X^*$ that separate points. For any topological space $Z$, show that a mapping $f:Z\to X$ ...
1
vote
1answer
53 views

Show that $\lim_{p \to \infty}||x||_p=||x||_M$ [duplicate]

Let $$||x||_p=\left( \displaystyle \sum_{i=1}^n|x_i|^p\right)^{\frac{1}{p}}$$ and $$||x||_M=\max\{|x_1|,|x_2|,...,|x_n|\},$$ norms in $\mathbb{R}^n$. Show that $$\lim_{p \to \infty}||x||_p=||x||_M, \ ...
1
vote
1answer
31 views

Prove scalar product in a normed vector space is an open mapping.

I have the feeling this is a really obvious question, but I'm having trouble with it, here it goes: Let $(X, ||\,||)$ be a normed vector space over $K$, prove that $\odot:K\setminus\{0\}\times X ...
0
votes
2answers
35 views

norm of canonical projection = 1

How do I show that given $M$ a closed subspace of a normed space $X$, and let $\pi$ be the canonical projection of X onto $X/M$. Prove that $\|\pi\| = 1$. I figure I could use Riesz' lemma and set ...
1
vote
2answers
47 views

a functional analysis question

$X$ is a banach space and $f$ a non zero linear functional. I'm trying to show $null(f)$ not dense in X $\implies f$ continuous. I've tried a few approaches but I think the following seems the most ...
0
votes
0answers
25 views

Is the set, $\{f\in Y: \left\|f\right\|_{Y} \leq C \ \text{and} \ \left\|(|f|)\right\|_{X}\leq C \}$, closed in $(Y, ||\cdot||_{Y})$?

Put, $X= \text{The space of "nice" complex valued functions on } \mathbb R; \text{that is}, f:\mathbb R \to \mathbb C;$ so that $X$ is Banach space with respect to the norm ...
4
votes
1answer
43 views

Duality mappings on finite-dimensional spaces

I have a few questions regarding some concepts in a book "nonlinear partial differential equations by Roubicek" I am studying. The following is from the text. "Let $V$ be a separable, reflexive ...
1
vote
1answer
43 views

Are $C^0[a,b]$ and $C^0[0,1]$ isometrically isomorphic?

Consider $C^0[a,b]$ and $C^0[0,1]$, each equipped with the $L^1$-Norm. Are these (out of curiosity) isometrically isomorphic?
1
vote
1answer
36 views

Cardinality of maximal subsets with some property

Let $X$ be a normed space. Is it true that all maximal (with respect to "$\subset$") subsets $D\subset X$ with the following property: $$ \|x-y\| \geq1 \textrm{ for } x\neq y, x,y\in D, $$ are of the ...
1
vote
1answer
30 views

Clarifying the PDE notation C^1([0,T], X).

In studying nonlinear hyperbolic PDE, I've come across the following spaces: $C([0,T],H^s(\mathbb{R}^n))$. $C^1([0,T], H^s(\mathbb{R}^n))$. $L^p([0,T],H^s(\mathbb{R}^n))$. I presume that $(1)$ ...
3
votes
1answer
72 views

Are $L_p$ spaces of functions with separable support separable?

Let $X$ be a separable space. Is $L_p$$(X, \mu, V)$ a separable space? Here, $(V, |\cdot|_V)$ is a normed space. And a norm of $L_p(X, \mu, V)$ is: $$ \|f\|=\left(\int_X \big(\vert f\rvert_V\big)^p ...
-1
votes
1answer
32 views

bounded subset of normed space

Suppose $X$ is a normed linear space and $S\subset X$. Show that if $$\sup_{x\in S}\{\mid f(x)\mid\}<\infty$$ for every $f\in X^{\ast}$, then $S$ is a bounded in norm.
0
votes
1answer
28 views

Inequality on Hilbert spaces in order to prove the nonexpansivity of a mapping.

I have an application $T\colon H\to H$ (where $H$ is a Hilbert space) such that $$(Tx-Ty,x-y)\leq \|x-y\|^2,\forall x,y\in H$$ where $(\cdot,\cdot)$ is the inner product of $H$ and $\|\cdot\|$ its ...
0
votes
3answers
53 views

Norm of vectors inequality

I tried proving this with triangular inequality but i was not right can any one help me with this
0
votes
2answers
89 views

Density of $C^{1}_{0}(\mathbb R)$ in $L^{\infty}(\mathbb R)$

I am looking for a counterexample to $C^{1}_{0}(\mathbb R)$ ( $C^1$ functions with compact support) is dense in $L^{\infty}(\mathbb R)$? Is there some easy counterexample showing that this latter is ...
2
votes
1answer
28 views

Norm of vector with respect to operator

Define $L$ is a linear operator maps from $E^n$ to $\mathbb{R}$, its norm is defined as $||L||_{op}=\sup\limits_{||x||=1}L(x)$, where $||\cdot||$ is any norm on $E^n$. How to show that ...
0
votes
1answer
50 views

The dual space of normed vector space $X$ is isomorphic to the dual of its completion

Let $\overline{X}$ be Banach space and $X$ be dense subset of it. Show that dual space of X and dual space of $\overline{X}$ are isomorphic. Why these are isomorphic? I don't know how to prove ...
0
votes
0answers
23 views

subjectivity of transpose and bounded from below

This is problem of Tao's epsilon of the room 1.5.13. Let $T : X \to Y $ be a continuous linear transformation which is bounded from below (i.e. there exists $c > 0$ such that $\|Tx\| \geq ...
1
vote
1answer
28 views

Compact embedding

Prove that the embedding $j\colon (C^1[0,1],\|\cdot\|)\to(L^1[0,1],\|\cdot\|_{L^1})$ where $\|f\|=\max\{\|f\|_\infty,\|f'\|_\infty\}$ and $\|f\|_\infty$ denotes the supremum norm, ...
0
votes
1answer
37 views

Continuous linear functional

I want to show that $f:(\ell^1,\parallel. \parallel_1)\to \mathbb K$ defined by $f((x_n))=\sum\limits_{n=1}^{\infty}\dfrac{\vert x_n\vert}{n}$ is continuous linear functional and the norm of $f$ is ...
2
votes
1answer
47 views

On the completeness of inner product spaces.

Let $H$ be a Hilbert space, equipped with an inner product $(\cdot,\cdot)_1$ and norm $\|\cdot\|_1$ induced by it. Let $(\cdot,\cdot)_2$ be other inner product on $H$ and $\|\cdot\|_2$ the norm ...
2
votes
2answers
56 views

Topology induced by Scalar Product

I just had an idea: It is clear that every scalar product induces a norm and that a metric and that finally a topology. Turning this argument around we know: Not every topology induces a metric only ...
0
votes
1answer
42 views

Is $H^2\cap H^1_0$ dense in $H_0^1$?

Let $-\infty<a<b<+\infty$ and $I=(a,b)$. Consider $H^1_0(I)$ equipped with the norm $\|\cdot\|_{H^1}$ given by $$\|f\|_{H^1}=\|f\|_{L^2}+\|f'\|_{L^2}.$$ Is $H^2(I)\cap H_0^1(I)$ dense in ...
0
votes
1answer
16 views

Range of a continuous linear mapping

I want to show that the range of the linear map $T:(\ell^1,\parallel .\parallel_1)\to (\ell^2,\parallel .\parallel_2)$ defined by $Tx=x$ is not closed. I considered a sequence $(x^{(n)})$ in ...
3
votes
1answer
79 views

Weak* continuity

Let $B$ be the open unit ball in $\mathbb{R}^2$ and $\mathcal{M}^+$ the set of nonnegative Radon measures on $B$ and $\mathcal{M}^2$ the set of $\mathbb{R}^2 \text{-valued}$ Radon measures on $B.$ I ...
1
vote
0answers
35 views

Weak convergence, weak neighborhoods

Let $V$ be a normed vector space, $V'$ its continuous dual. Let $U \subset V$. Consider the statements: i) For any finite $F \subset V'$ there exists $y \in U$ with $\max_{f \in F} |f(y)| < 1$. ...
3
votes
2answers
40 views

Why is the image of a compact operator separable?

Let $A$ and $B$ be normed vector spaces and let $S\in \mathscr{K}(A,B)$ be a compact operator. Question: How does it follow that the image of $S$ is separable? Thanks for the help.
2
votes
4answers
111 views

Why is $C_c^\infty(\Omega)$ not a normed space?

I am watching a Coursera video on Théorie des Distributions and I am trying to understand one of the slides. Let $\Omega \subset \mathbb{R}^N$ be an open set and $C_K^\infty(\Omega) = \{ \phi \in ...
2
votes
1answer
45 views

Show that E\H (Hyperplane) is arc-connected $\Longleftrightarrow$ H isn't a closed subspace

Good morning, Let $E$ be a real normed vector space and $H$ a hyperplane of $E$ Show that E\H is arc-connected $\Longleftrightarrow$ H isn't a closed subspace I have no idea to solve it. But If $f$ ...
2
votes
0answers
58 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
votes
1answer
97 views

If the dual spaces are isometrically isomorphic are the spaces isomorphic?

Let $X$, $Y$ be Banach spaces such that the duals $X^\ast$ and $Y^\ast$ are isometrically isomorphic. Are $X$ and $Y$ necessarily isomorphic? The answer to the question whether $X$ and $Y$ are ...
1
vote
1answer
48 views

Preannihilator of the image of an adjoint of a bounded operator

Let $E,F$ be normed spaces and $F\colon E\rightarrow F$ be a linear bounded operator. Denote by $$A'\colon F'\rightarrow E'$$ the adjoint of the operator between the topological duals of the normed ...
1
vote
2answers
87 views

Collecting things that are preserved by (isometric) isomorphisms between normed spaces

I would like to collect a list of things that are preserved under isomorphisms and isometric isomorphisms. The reason is that I hope to get a better perception of their importance in Functional ...
2
votes
1answer
59 views

Convergence in Sobolev Spaces

Consider the bounded mapping $A:W^{1,p}(\Omega) \rightarrow W^{1,p}(\Omega)^{*}$ where $A$ is defined as: $\langle A(u),v \rangle\text{ } := \int_{\Omega}a(x,u,\nabla u)\cdot \nabla v + c(x,u,\nabla ...
0
votes
0answers
76 views

Absolutely convergent series in normed linear space

I want to prove that in a normed linear space $X$ if for all absolutely convergent series $\sum\limits^{\infty}_{n=1}x_n$, the series $\sum\limits^{\infty}_{n=1}T(x_n)$ is convergent, then $T:X\to Y$ ...
1
vote
0answers
66 views

Every inner product space is a normed space which is also a metric space

As I am pretty sure that everybody knows that a Hilbert space is a space that is a complete, separable and generally infinite dimensional inner product space. By the means of completion, every Cauchy ...
2
votes
1answer
69 views

restriction a non compact operator to compact operator

If $T\in\mathcal{B}(X,Y)$ is not compact can the restriction of $T$ to an infinite dimensional subspace of $X$ be compact?
3
votes
1answer
95 views

Weak continuity in Sobolev Spaces

First consider the following two Sobolev Embedding Theorems. Theorem 1: The continuous embedding $W^{1,p}(\Omega) \subset L^{p^{*}}(\Omega)$ holds provided the exponent $p^{*}$ is defined as ...
0
votes
1answer
88 views

Norm of Matrix transpose

I have a problem below: Let $\|\cdot\|$ denotes the norm matrix \begin{equation} \|A\|=\max \frac {\|Ax\|}{\|x\|}, \end{equation} for every $A$. Now suppose that $H: \mathbb{R}^k \rightarrow ...
1
vote
1answer
46 views

About the closedness of Banach space

I'm reading a proof in trying to prove that a Banach space $X$ is reflexive if and only if $X^{*}$ is reflexive. There's a point in the proof saying that $X$ is a closed subspace of $X^{**}$, but I ...
2
votes
0answers
60 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 ...