4
votes
2answers
51 views

Riesz's Lemma for $l^\infty$ and $\alpha = 1$

Riesz's Lemma says the following: Let $X$ be a normed vector space and $Y$ a proper closed subspace of $X$. Pick $\alpha \in (0,1)$. Then $\exists x\in X$ such that $|x|=1$ and $d(x,y) \geq \alpha$ ...
-1
votes
0answers
22 views

Get locally uniformly convex norm by bounded linear operator

I want to prove this theorem Let $Y$ be a locally uniformly convex Banach space and $X$ be a Banach space and let $T$ be a bunded linear operator from $X$ into $Y$ such that for every bounded ...
1
vote
1answer
32 views

Proof of compactness for sets of norm equal to one in finite-dimensional normed vector spaces

The proposition I have been trying to prove is that the set $A=\{x\in E:N(x)=1\}$ is a compact subset of the (real) finite-dimensional vector space $E$ for any norm $N:E\to \mathbb{R}$. I am reading ...
6
votes
1answer
118 views

Radon-Riesz & Kadec-Klee

Let us say that a normed vector space has the a) RR (Radon-Riesz) property if for any sequence, norm convergence is equivalent to weak convergence + convergence of norms. b) KK (Kadec-Klee) property ...
1
vote
1answer
45 views

Spectrum of a finite rank operator

If $ T\in B(H)$ is a finite rank operator, then there are orthonormal vectors $e_1,...,e_n$ and vectors $g_1,...,g_n$ such that $Th=\sum_{i=1}^n (h,e_i )g_i$, then we can easily see that $T$ is ...
0
votes
2answers
52 views

continuity in the strong topology implies continuity in the weak one

I have to prove that if $T:(E,\|\cdot\|_E)\rightarrow (F,\|\cdot\|_F)$ is a continuous and linear operator, and $x_h\rightharpoonup x$ in $E$, than $Tx_h\rightharpoonup Tx$ in $F$. So we know that $T$ ...
2
votes
0answers
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 ...
3
votes
0answers
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
votes
1answer
21 views

convex weak* sequentially closed subset of a separable Banach space implies weak* closed

I'm studying Conway's a course in Functional Analysis by myself. The following is corollary 6.12.7 of this book. If $X$ is a separable Banach space and $A$ is a convex subset of $X^*$ that is weak* ...
2
votes
0answers
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 ...
0
votes
0answers
47 views

Show that an hyperplane is closed iff f is linear and continuous

I need an help with the following exercise. Let $(E,\| \cdot \|)$ a n.v.s. and let $f:E\rightarrow \Bbb R$. Show that $H=\{x\in E: f(x)=\alpha\}$ is closed if and only if $f\in E'.$ Actually, I ...
0
votes
1answer
32 views

Show that an operator is bounded.

Let $\{\alpha_{mn} ;m,n\geq 1\}$ be scalars satisfying a- $M=\sup_n\sum_{m\geq 1}|\alpha_{mn}|<\infty $ , and b- $\sup_n|\alpha_{mn}|<\infty$, then $(Af)(n) = \sum_{m\geq 1}\alpha_{mn} f(m)$ ...
5
votes
1answer
61 views

When is $M+N$ closed

Let $X$ be a Banach space and $M,N$ be closed subspaces. If the range of linear transformation $x\to (x+M)\oplus (x+N)$ from $X$ into $X/M\oplus X/N$ is closed show that $M+N$ is closed. or using ...
0
votes
0answers
16 views

Converse to sequential Banach--Alaoglu [duplicate]

Let $B$ be the closed unit ball of the dual space of a real normed vector space $V$. If $V$ separable then $B$ is sequentially compact in the weak-* topology. What about the converse?
4
votes
1answer
100 views

Reproducing Kernel Hilbert Space (RKHS) constructed by the summation of positive-definite kernels.

Let $k_1,\ldots,k_p$ be positive definite kernels defined on $\mathcal{X}\times\mathcal{X}$, where $\mathcal{X}$ is a non-empty set. $k_i$ is the reproducing kernel of the Reproducing Kernel Hilbert ...
0
votes
0answers
14 views

Quotient space and continuous linear operator

I'm trying to study some arguments of math by myself and I have some problems to understand the interpretation of the norm about linear operators. The books says that there's a correspondence between ...
1
vote
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} ...
0
votes
0answers
19 views

Minimization of an evaluation under the weak* topology

I'm self studying (for fun) the book "Functional Analysis, Calculus of Variations and Optimal Control" (by Clarke), and I'd appreciate some feedback for my proposed solution for an exercise from the ...
2
votes
1answer
52 views

Series in a space which is not complete

Let $X$ be a normed vector space and $\left\lbrace x_n \right\rbrace_{n \in \mathbb{N}} \in X^{\mathbb{N}}$ with $$\sum_{n=1}^{\infty} \|x_n\| < \infty \wedge \sum_{n=1}^{\infty} x_n \notin X,$$ ...
0
votes
2answers
52 views

Is my proof correct? Finite-dimensional normed vector spaces

I'm trying to prove that every finite-dimensional normed space is topological isomorphic to $\mathbb{R}^n$. Let $(E,\|\cdot\|_E)$ such that $dimE=n$ and let $$ T:\mathbb{R}^n\to E\\ x\mapsto ...
0
votes
1answer
28 views

Continuity of a map to a Frechet space

Let $(A,\| \cdot \|)$ be a normed space and $B$ be a Frechet space equipped with a family $\{ p_k \}_{k \in \mathbb{N}}$ of seminorms. Let $\phi: A \to B$ be a linear transformation satisfying the ...
2
votes
0answers
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
votes
1answer
50 views

Implying a positive definite operator

If we are given that $A:V \rightarrow V$ is an operator where $V$ is a real Hilbert space. If we are given that $A$ is bounded, strictly positive $\big(\langle Au,u \rangle > 0$ for all $u \neq ...
2
votes
1answer
61 views

Finite-dimensional subspace normed vector space is closed

I know that probably this question has already been answered, but I'd like to present my attempt of solution. Let $(E,\|\cdot\|)$ be a normed vector space and let $F\subseteq E$ be a ...
2
votes
1answer
62 views

How can I prove that $f$ is continuous at $0$?

Let $E$ and $F$ linear normed spaces, and consider a linear function $f:E\to F$ which satisfies for all $(x_n)\in E^{\mathbb{N}}$ such that $x_n\to 0$ then $f(x_n)$ is bounded in $F$. Then I have to ...
4
votes
1answer
54 views

Surjection of norms

Let $V$ be an infinite dimensional $\mathbb{C}$ (or $\mathbb{R}$) vector space. Suppose there exists two norms on $V$ such that \begin{equation*} \| \cdot\|_1 \leq \| \cdot \|_2. \end{equation*} Is ...
1
vote
1answer
36 views

Exists $l\in X^*$ such that $\|l\|=1, l|_Y=0$ and $l(x_0)=\operatorname{dist}(x_0, Y)$?

Let $(X, \|.\|)$ be a normed space and $Y\subsetneq X$. How can you prove that for $x_0 \in X\setminus Y$ there exists $l\in X^*$ such that $\|l\|=1$, $l|_Y=0$ and $l(x_0)=\operatorname{dist}(x_0, Y)$ ...
1
vote
1answer
40 views

The continuous dual of the reals

I just have a few questions involving the continuous dual of $\mathbb{R}^{N}$. We know that the dual $(\mathbb{R}^{N})^{*}$ of $\mathbb{R}^{N}$ is the space of all linear forms $$a: \mathbb{R}^{N} ...
0
votes
1answer
37 views

Theorems on continuous embedding

Let $X,Y$ be two normed spaces such that $X\hookrightarrow Y$. Show the following: 1) If $A\subset X$ is closed in $Y$ then $A$ is closed in $X$; 2) If $X, Y$ are Banach spaces if $x_n$ tends ...
0
votes
0answers
30 views

Weakly compact closed balls in reflexive space

If it is given that $X$ is a reflexive Banach space. Let $K \subset X$ be a norm closed and norm bounded convex set. I want to show that $K$ is weakly compact. I have the following idea but I am not ...
3
votes
0answers
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 ...
1
vote
1answer
36 views

Convergence of a sequence of unit vectors in a Hilbert space

Let $E$ be a vector space (over $\mathbb{R}$) with a positive definite hermitian form and let $\{x_{n}\}$ ($x_{n} \not= 0$) be a sequence converging to $x$ in the $L^{2}$ norm ...
1
vote
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 ...
0
votes
1answer
25 views

difference of Cauchy sequences is Cauchy

Let $X$ be a normed space and let $(x_n)$ and $(y_n)$ be Cauchy sequences on $X$. Show that $z_n=\mid\mid x_n-y_n\mid\mid$ is also a Cauchy sequence on $\mathbb{R}.$ My answer: $ | z_n-z_m | = | \ \ ...
0
votes
1answer
31 views

The modulus of smoothness of $c_0$ by an equivalent norm

Let $(X,\|\cdot\|)$ be a Banach space. For $t>0$, the modulus of smoothness of $\|\cdot\|$ is defined by $\rho_X(t)=\sup\left\{\dfrac{\|x+ty\|+\|x−ty\|}{2}−1:x,y\in S_X\right\}$. We define an ...
1
vote
1answer
30 views

the modulus of smoothness of Hilbert space

Let $(X, ||.||)$ be a Banach space. For $t>0$, the modulus of smoothness of $||.||$ is defined by $$\rho_X(t)=\sup\left\{\frac{||x+ty||+||x-ty||}{2}-1: x,y\in S_X\right\}.$$ Ican easily calculate ...
2
votes
1answer
54 views

Is continuous extension on dense subset an isometry

If we have that $X \subset V$ is dense linear subspace. Where $V$ is normed space. I can show that for any $f \in X^{*}$, there exists a unique extension $\bar{f}$. I want to know if it can be shown ...
0
votes
1answer
36 views

Existence of nontrivial bounded linear operator?

Are there two normed spaces such that there is no nontrivial bounded linear operator between them: $$\nexists T:X\to Y: T\text{ nontrivial, linear and bounded}$$
0
votes
1answer
69 views

Equivalence of norms in finite-dimensional spaces

Assume we have a separable, reflexive Banach space $X$ such that $\{e_{i}\}$ is a basis. Let $X_{n} = \text{span}\{e_{1},\ldots,e_{n}\}$ be finite-dimensional subspaces where we define members ...
2
votes
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 ...
0
votes
1answer
34 views

Some questions about subspaces in Banach spaces

I just have a few question about some things in Banach spaces. Let $X$ be a separable, reflexive Banach space with basis $\{e_{i}\}$. Let $X_{n} = \text{span}\{e_{1},...,e_{n}\}$, then consider the ...
-1
votes
1answer
44 views

Norm Space verification

For each of the following decide whether the suggested formula defines a norm on the indicated space. You may assume that $||f||_1=\int_0^1 |f(t)| dt$ does give a norm on the space of all continuous ...
0
votes
0answers
37 views

Convergent sequences in normed vector spaces

Consider a normed vector space $X$. I have a couple of questions regarding convergent sequences and subspaces of $X$(topological subspaces, not necessarily linear subspaces). Let $W$ be a topological ...
2
votes
0answers
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 $z∈X$ ...
0
votes
0answers
55 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
votes
0answers
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 ...
0
votes
1answer
45 views

What is the norm of this linear functional?

Let $X = C[0,1]$ be the space of continuous functions with the max norm. Then define the linear functional; $$f(x) = \int_0^1 x(t)dt - x(1/2)$$ Then is the norm of this operator 1 or 0? I'm a little ...
0
votes
0answers
39 views

geometry interpretation of uniformly rotund in every direction

The norm of a Banach space $X$ is said to be uniformly rotund in every direction if $\lim_{n\to\infty} ||x_n-y_n||=0$ whenever $x_n, y_n \in S_X$ are such that $\lim_{n\to\infty} ||x_n+y_n||=2$ and ...
1
vote
1answer
57 views

Weak convergence on Banach space

Let $(X,|.|)$ be a Banach space, and $Y$ is a linear subspace of $X$ which is dense in $X$. Now if we have another norm $|.|_Y$ in $Y$ which is comparable to $|.|$ by $$|y|\leq C |y|_Y, \ \ \ \forall ...
4
votes
3answers
91 views

Showing that a sequence converges in norm

I am trying to prove the following: Let $X$ be a normed linear space satisfying the property: $\forall \left\{x_n\right\}, \left\{y_n\right\} \subseteq X $, we have $\|x_n\|=\|y_n\|=1, ...