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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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?
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51 views

Alternatives to the notation $\|x\|$ for the norm of $x$?

For aesthetic reasons, I don't like the notation $\|x\|$ for the norm of $x$. Have any alternatives been proposed?
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112 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 ...
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3answers
70 views

Relation between continuous maps and convergence of sequences

I am studying metric spaces and I know that in a normed space $E$ a map $T:E \to E$ is contínuous if and only if $T(x_n) \to T(x)$ for every convergent sequence $x_n \to x$ in $E$. In my notes there ...
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1answer
42 views

Isometry in a finite dimensional vector space is always surjective

My book defines an isometry as a linear operator between two vector spaces X and Y where: $$\|T(x)\|=\|x\|$$ Later it has a sentence which I do not understand. If we have a finite dimensional ...
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15 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 ...
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50 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} ...
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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 ...
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3answers
29 views

Seminorm proof of a function

I have an example in a book which is not very clear to me : let $E$ vector space made of numerical functions (or complex) $f$ defined on a set $A$. $\forall a \in A, N_a : f \rightarrow |f(a)|$ is a ...
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1answer
53 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 $$\sum_{n=1}^{\infty} \|x_n\| < \infty \wedge \sum_{n=1}^{\infty} x_n \notin X,$$ ...
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45 views

Finding a homeomorphism between these two balls

Let $u_1,u_2,u_3 \in \Bbb C$ be the cubic roots of unity. Define two norms on $\mathbb{C}^2$, $$\Vert (x,y) \Vert_1 = \sqrt{\vert x \vert^2 +\vert y \vert^2} \ \text{and} \ \Vert (x,y) \Vert_2 = ...
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2answers
54 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 ...
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1answer
29 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 ...
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1answer
89 views

Uniform convex space

Please I want to know if this space $$H^1_{0,p}([0,+\infty))=\lbrace u, u\in AC([0,+\infty)), u(0)=u(+\infty)=0,\sqrt{p}u'\in L^2\rbrace$$ where $p>0$, $p\in L^1((0,+\infty))$ ...
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0answers
32 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 ...
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1answer
25 views

Equivalence of sets

Let $u_1, u_2, u_3 \in \mathbb{C}$ be the cubic roots of unity I'm wondering if the following two sets (balls) are equivalent: $$ \lbrace (v,w) \in \Bbb C^2 : \vert v \vert + \vert w \vert \leq 1 ...
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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 ...
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15 views

Distance of an element to $\ker f$ in a normed vector space. [duplicate]

Let $E$ a normed vector space and the hyperplane $H=\ker f$ with $f\in \mathcal{L}(E, \mathbb{R})$. Prove that if $a\in E$ then $\displaystyle{d(a, H)=\frac{|f(a)|}{\|f\|}}$
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1answer
79 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 ...
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1answer
63 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 ...
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0answers
16 views

H2 norm or H2 control probloem

I read a paper, it says: How to understand this definition of H2? I know the definition of the H2-norm of a system: or However, how to understand the paper one's from these formula or ...
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2answers
32 views

Express Norm Using Inner Product

I'd like to know whether there's a way to express a norm using inner product, for example , is there any inner product we may use that is equal to $(||Ax-b||_2)^2$ ? Thanks in advance.
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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 ...
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26 views

Which of the following defines norm on V?

Let $V$ denote the vector space of all polynomials over $\mathbb R$ of degree less than or equal to $n$. Which of the following defines a norm on $V$. $\|p \|^2 = |p(1)|^2 + \cdots + |p(n+1)|^2, p ...
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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 $l(x_0)=\operatorname{dist}(x_0, Y)$ ...
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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} ...
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1answer
38 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 ...
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1answer
31 views

What truly are length, area and volume? And considerations about divergence in normed spaces

All the "(?)" are parts when i'm not sure at all if what i'm saying is right or not, it's just my intuition. Part 1 In $\mathbb{R}$, we can define the length of a segment. In $\mathbb{R}^2$, the ...
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39 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 ...
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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 ...
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1answer
40 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 ...
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0answers
36 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 ...
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1answer
27 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 | = | \ \ ...
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1answer
15 views

I need help showing something is a linear continuous operator.

Define $T:C([0,1])\rightarrow C([0,1])$ by $T(f)(x)=f(0)+\int_0^xtf(t)dt$ I want to show that $T$ is a continuous linear operator.Showing the linear part is easy enough, but I am not quite sure how to ...
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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 ...
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1answer
32 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 ...
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1answer
59 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 ...
3
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2answers
70 views

Where is the error in my proof?

I have this excercise. I am able to solve it, but the problem is that I can solve it without using the last part of information of the existence of the u-vector. That makes me afraid that my proof is ...
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1answer
37 views

Existence of nontrivial bounded linear operator?

Are there two normed spaces such that there is no nontrivial bounded linear operator between them: $$\nexists T:X\to Y: T\text{ nontrivial, linear and bounded}$$
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1answer
73 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 ...
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3answers
63 views

Let $V $be a vector space. Prove/Disprove: There is a norm $\|\cdot\|$, such that all subsets of $V$ are open sets in $(V,\|\cdot\|)$.

The Assignment: Let $V$ be a vector space over $\mathbb{R}$ with $V \not= \{0\}$. Prove or disprove: There is a norm $\|\cdot\|_d$ on $V$, such that all subsets of $V$ are open sets in ...
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27 views

Show that $f: K_1(0) \rightarrow \mathbb{R}^3$ is Lipschitz.

Firstly, the Assignment: Let $V = (\mathbb{R}^3 ,\|\cdot\|_{\infty})$ where $\|\cdot\|_{\infty}$ denotes the maximum norm and consider the function: $$f: K_1(0) \rightarrow ...
2
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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 ...
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1answer
41 views

Comparing two linear functions

Let $X$ be a Banach space and let $h:X\to\Bbb C$ and $f:X\to\Bbb C$ be two bounded linear functions such that if for some $x\in X$ we have $f(x)=0$ then $h(x)=0$. Prove that there exists a ...
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1answer
35 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 ...
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1answer
37 views

The canonical surjection between the full and the reduced group C^*-algebras

This might be an incredible easy question -- since any reference I've found state it as obvious -- but anyway: Given a group $G$, I can construct the full group-$C^*$-algebra $C^*(G)$ be completing ...
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1answer
46 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 ...
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39 views

can arbitrary norm in $R^n$ be uniformly continuous?

let $d_1$ be an arbitrary norm on $\mathbb R^n$, and let $d_2$ be a max-norm on $\mathbb R^n$ (max-norm : http://en.wikipedia.org/wiki/Maximum_norm) How can I show that $d_1$ : ($\mathbb R^n$, $d_2$) ...
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0answers
19 views

my question is about proving a norm is a norm

I want to show that $||f||=\max_{t\in [0,1]}|f(t)|$ defines a norm on $C([0,1])$. I only have a question on the triangle inequality property. This is what I have done but I am not quite sure it is ...
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0answers
39 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 ...