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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23 views

Proof of “Dual normed vector space is complete”

http://en.wikipedia.org/wiki/Dual_norm As in the introduction of dual norm by Wiki, it says dual normed space $X'$ is always complete. How to prove that? or at least explain that? We all know the ...
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48 views

In $\mathbb{R}^{n}$ all norms are equivalent

While trying to prove the Theorem mentioned in the Title, I got stuck in the inequality shown below. I think that the proof uses the $\epsilon$ and $\delta$ definition of continuity but I am not ...
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3answers
40 views

Equivalency of Norms and the Open Mapping Theorem

"Let $\|\cdot\|_1$ and $\|\cdot\|_2$ be two norms on the space $X$ s.t. $(X,\|\cdot\|_1)$ and $(X,\|\cdot\|_2)$ are both complete. Assume that for any sequence $(x_n) \subseteq X$, $\|x_n\|_1 \to 0$ ...
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34 views

Completing the solution, lipschitz maps inducing other maps

Let $(X, d)$ be a metric space, $(E, || \cdot ||)$ a Banach space, $(AE(X), || \cdot ||)$ - as described below. I've already proven that for any Lipschitz function $u: X \rightarrow E $ there exists ...
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1answer
20 views

equivalnce of linear functions, which one's kernel includes the other's

The following is from my homework. PLEASE don't reveal all the solution, but leave at least something for my imagination. Let $X$ be a normed space. Let $\phi,\psi : X → \mathbb C$ be linear ...
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1answer
54 views

Applications of Stone-Weierstrass Theorem

For every continuous function $f:[0,1]\rightarrow \mathbb{R}$, prove that there exists a sequence of polynomials $p_n$ such that $p_n$ converges to $f$ on $[0,1]$ and for every $x\in [0,1]$, we have ...
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1answer
19 views

Completeness of normed vector spaces

Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be continuous such that $f$ vanishes at infinity. i.e. for all $\epsilon >0$, there exists $N\in \mathbb{N}$ such that $n>N$ implies ...
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1answer
31 views

Function sequences in $C[0,1]$ using infinity norm

I am working on the space of continuious function from $[0,1]$ to $\mathbb R$ with the infinity norm ($ \sup_{x\in [0,1]}|f(x)|$). My question is the following Is it possible to construct a sequences ...
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69 views

Norm defined by matrix

Suppose $A$ is an $n\times n$ real matrix and for $\mathbf{x} \in \mathbb{R}^n$ define $\|\mathbf{x}\|_A = \sqrt{\mathbf{x}^T A \mathbf{x}}$. Under what conditions on the matrix $A$ is $\|\cdot ...
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28 views

Scalar product of $L_2$ with $\mu(E):=\int_E gdx$

I read in Kolmogorov and Fomin's Элементы теории функций и функционального анализа (p. 400 here) that, if we define measure $\mu$ for $E\subset[-1,1]$ by $$\mu(E):=\int_E g(x)dx$$ where the integral ...
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28 views

How to show that $F(x)$ continuous?

$F:\Big(C[0,1],||.||_2\Big)\rightarrow \Big(C[0,1],||.||_3\Big)$ $x\rightarrow F(x)(t)=\int^t_0x(s)ds,\quad\quad0 \le t\le 1 $ Show that F is continuous. F is linear. for n=0,1,2.. $x_n(t)=t^n,0 ...
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0answers
25 views

Continuous functions dense in $L_p(X,\mu)$ if $X$ has a special property

Let $X$ be a metric space endowed with a measure $\mu$ satisfying the following condition: all the open and closed sets of $X$ are measurable and for any measurable set $M\subset X$ and any ...
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1answer
55 views

Approximation of $f\in L_p$ with simple function $f_n\in L_p$

Let us use the definition of Lebesgue integral on $X,\mu(X)<\infty$ as the limit$$\int_X fd\mu:=\lim_{n\to\infty}\int_Xf_nd\mu=\lim_{n\to\infty}\sum_{k=1}^\infty y_{n,k}\mu(A_{n,k})$$where ...
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1answer
84 views

Best approximation for a closed set in a finite dimensional normed space

First of all I'd like to mention that it is a part of my home work so I'd like if you won't give the answer itself, but try to guide me into it. I've been losing my mind for the last couple of hours ...
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1answer
30 views

$L_1\subset L_p$?

I am trying to check whether the implication $\forall p>1\quad f\in L_p(X,\mu)\Rightarrow f\in L_1(X,\mu)$ is true when $\mu(X)<\infty$. By $L_p(X,\mu)$ I mean the space of Lebesgue integrable ...
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2answers
29 views

Showing Convergence in $L^p$ norms

Let $X$ be a finite measure space and $1\le p<\infty$ and $\{f_n\}$ be a sequence in $L^p(X)$ such that coverge to $f$ in $L^p(X)$ . If there exists constant $K$ such that for every $n\in ...
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0answers
21 views

Subspaces of same finite codimension are isomorphic

I would like to show that two subspace $Y$ and $Z$ of a normed space $X$ are isomorphic provided $\text{codim } Y = \text{codim } Z <\infty$. I can show that $\text{codim}(Y\cap Z) <\infty$ but ...
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0answers
28 views

Metric space with measure and a special property

Let $R$ be a metric space endowed with a (complete) measure $\mu$ satisfying the following condition: all the open and closed sets of $R$ are measurable and for any measurable set $M\subset R$ and any ...
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1answer
28 views

Does this funcion define a norm on $\mathbb{C}^n$?

Let $m$ and $n$ be two given positive integers. And, let $f \colon \mathbb{C}^n \to \mathbb{R}$ be defined as follows: $$ f(x_1, x_2, \ldots, x_n) \colon= \left( \sum_{i=1}^n \sqrt[m]{|x_i|} ...
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0answers
39 views

Does Hlawka Inequality follow from Triangle Inequality?

On MathOverflow I saw this inequality. Let $E$ is a normed linear space. $$ \|x+y\|+\|y+z\|+\|z+x\|\le\|x\|+\|y\|+\|z\|+\|x+y+z\|,\qquad\forall x,y,z\in E $$ Apparently this is always true if $E = ...
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26 views

Are there properties of vector space equipped with two norms?

I am interested in a vector space equipped with two norms$ \lvert \lvert \cdot \rvert \rvert$ and $ \lvert \lvert \cdot \rvert \rvert ^*$ satisfies that there is $M>0$ such that $ \lvert \lvert x ...
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12 views

Bounded Linear Maps on Normed Vector Spaces

Let $A$ be an $m\times n$ matrix $(\alpha_{jk};\;j=1,...m,k=1,...,n).$ As we know, $$[Bx]_j = \sum_{k=1}^n\alpha_{jk}x_k,\;\;\;\;\;j=1,...,m,\;\;\;x=(x_1,...,x_n),$$ defines a bounded linear operator ...
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1answer
35 views

Convergence of a sequence of linearly independent vectors in normed space

In an infinite dimensional normed vector space is it possible to find a sequence ${v_n}$ of linearly independent vector (so the sequence is a set of linearly independent vectors) each has norm 1 such ...
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1answer
44 views

distance between solutions in a convex optimization

Assume that you have the following convex optimization problem: $\min_{M} \|b+A\ M\ v\|_2$ subject to : $\|M\|_{2}<1$ (maximum singular value less than 1) where M is a suare matrix (n by n), A ...
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1answer
28 views

$ {\|f\|}_p = \sqrt[p]{\int_{a}^{b} |f(x)|^p {\rm d}x}$ is a norm

Consider the space $C([a,b])$ of all continuous functions $f\colon [a,b]\rightarrow \mathbb{R}.$ Show that the function $\|\cdot\|_p\colon C([a,b]) \rightarrow [0,\infty),p>1$, given by $$ ...
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1answer
24 views

Is $\phi$ a norm of E?

Let $(E, \| \|)$ be a normed space. We define $\phi:E \rightarrow [0,\infty)$ as follows: $$\phi(e)= \dfrac{\|e\|}{1+\|e\|}$$ Is $\phi$ a norm of $E$? Please help! Thank you! P.S. This question ...
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323 views

Non-Banach, completely metrizable normed vector space

Does there exist a normed vector space $(X,\|\cdot\|)$ over $\mathbb R$ or $\mathbb C$ such that the metric induced by the norm $(x,y)\mapsto\|x-y\|$ is not complete; but there exists some other ...
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1answer
28 views

What is the generalized form of this identity and how to interpret it?

I have learnt that for any inner product space of $\mathbb{C}$, we have $$\langle f,g\rangle=\frac{1}{4}\Big[||f+g||^2-||f-g||^2+i\big(||f+ig||^2-||f-ig||^2\big) \Big]$$ I know how to prove it, but ...
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2answers
60 views

On Equivalent Norms in an Infinite Dimensional Vector Space

How many non-equivalent norms can we define in an infinite dimensional vector space? Is there any explicit expression?
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1answer
32 views

When a metric space is a normed space?

I'm trying to figure out that which condition should be provided for a metric space to be normed also?
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20 views

Separable $X^*$ Property

Let $X$ be a normed space. If $X^*$ is separable, then there exists $(f_n)_{n\geq1}\subset X^*$ such that $\|f_n\|_{X^*}=1$ for all $n$ and $\{f_n:n\geq1\}$ is dense in $\{f\in X^*:\|f\|=1\}$. In ...
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0answers
26 views

Isometric isomorphisms between normed spaces and compact hausdorff spaces

Let $X$ be a normed space. Show that there is a compact Hausdorff space $Y$ such that $X$ is isometrically isomorphic to a subspace of $C(Y)$. I think this might be proved using the Banach–Alaoglu ...
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1answer
25 views

Compute the following norm derivatives

I was wondering if anyone can explain me how to compute the derivatives of the following norms: $\frac{d}{ds}||x+sp||^2_q$ for $x,p\in\mathbb{R^n}$ and $1<q<\infty$ $\bigtriangledown ...
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1answer
28 views

Are the normed spaces $ \mathbb{R}^{n^2} $ and $ M_n(\mathbb{R}) $ isometric?

Consider the spaces $ \mathbb{R}^{n^2} $ with euclidean norm and $ M_n(\mathbb{R}) $ of $n\times n$ matrices with the norm defined by $ \Vert A\Vert = \sup\limits_{\Vert x\Vert \le 1}\Vert Ax\Vert$. ...
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0answers
39 views

Normed Space and Hibert Space Problem

Anyone could describe me, why this is True? Suppose $(H, \|.\|) $ is a normed space. the norm $\|.\|$ induced by an inner product if and only if Parallelogram law is valid. Regards.
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0answers
52 views

Can a Norm be Induced by two Different Complex Inner Products?

Let $(X,\|\cdot\|)$ be a normed vector space over $\mathbb{C}$. If $\|x\|=\sqrt{\langle x,x\rangle}$ and $\|x\|=\sqrt{\langle x,x\rangle'}$ for all $x\in X$ where $\langle,\rangle$ and ...
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0answers
33 views

Normed space of dimension $> 1$

Let $X$ be a normed space of dimension > 1. Prove that each $\epsilon > 0$ exist $x,y \in X$ such that $\left \| x \right \| = \left \| y \right \| = 1 $ and $ 0< \left \| x-y \right \|< ...
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1answer
39 views

normed space of infinite dimension

By $X$ we denote an infinite-dimensional normed space. Show then that exist $x \in X$ and a closed subset $F\subset X$ such that $d(x,F) < \left \| x - y \right \|$ for all $y \in F$. (Note: It ...
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3answers
48 views

A direct proof on if $(X, ||\cdot ||)$ is a normed vector space and $Y\subset X$, with $Y$ having finite dimension, then $Y$ is closed.

I am trying to produce a direct proof on the statement mentioned above. The field I am working in is $\mathbb{R}$. My proof outline goes as following: If $Y$ is finite-dimensional, there exists a ...
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18 views

Normed spaces welfare problem

A producer own a technology for transforming an input into an output. If x $\in$ $\mathbb{R}_+$ units of the input are employed, the technology yields at most an amount √x ∈ R+ of output. Let F := ...
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2answers
33 views

Convex hull, compactness, normed spaces

Let $(X,\| \cdot \|)$ be a finite dimensional normed space. Show that if $S\subseteq X$ is compact, then the $\text{Conv(S)}$ is also compact. I used the Caratheodory's theorem to show that ...
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1answer
23 views

Is there a bounded dense subset of norm linear space?

I have a question. In norm linear space $X$, we can find a bounded dense subset of $X$, can´t we?
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1answer
36 views

Normed space and convex hull of closed subset

Let $(V, ||\cdot||)$ be a normed space. If $ C\subseteq V$ is a closed set we do not know if $ch(C)$ is closed or not. The professor provided this example that as of now I'm not getting: Consider the ...
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1answer
43 views

Some Strange Minimum and proof

I read following on Norm chapter in one book. $$\begin{align}\left|\left\|x-y\right\|-\left\|w-z\right\|\right| \leq & \min \{\|x-w\| + \|y-z\| , \|x-z\| + \|y-w\|\}\\\text{ or, }&\min ...
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1answer
50 views

Space of real polynomials of one variable isn't complete

Consider $E$ - the vector space of all real polynomials of one variable. I need to prove that it is not complete under any norm. I was thinking I could use the fact that certain functions, for ...
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1answer
136 views

Why isn't $\,\mathcal C[0,1]$ a Banach space in this unusual norm?

I wish to ask the following question: Let $\mathcal X$ be the normed space $\,\mathcal X=\mathcal C([0,1])$, with norm defined as $$ \|\,f\|= \max_{x\in[0,1]} x^2 \lvert\,f(x)\rvert. $$ Why isn't ...
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2answers
36 views

A General Question arises about norms and limits [closed]

Suppose $x_n \to x_0$, i.e. $\lim x_n=x_0$. How we can show that $||x_n|| \to ||x_0||$, that is $\lim_{n \to \infty} ||x_n||=||x_0||$, so $\lim_{n \to \infty} ||x_n|| = ||\lim_{x \to \infty} x_n||$?
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1answer
35 views

Norms for which every subset of closed unit ball containing the open unit ball is convex

It can be shown without much difficulty that any Euclidean norms satisfies the following condition :$$(P) \quad B \subset X \subset B' \Rightarrow X \, \text{is convex}$$ where $B=\{x \in E / \|x\| ...
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1answer
36 views

Continuity and norm of a functional

Let $E = \mathbb{R} [X]$ equipped with the norm $||p|| = \int_0^1 (|p(t)| + |p'(t)|) \ d t $. Check if the functional $\psi : E \ni p \rightarrow p(0) \in \mathbb{R}$ is continuous, and if it is, ...
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1answer
34 views

Prove some Equivalences Norm

Suppose $X=R^2$ and $x=(x_1, x_2)$. I see the following are equal EDIT: ( equivalence). why? i couldent find any proof to satisfy me. any hint or idea or proof highly appreciated. $||x||_1= |x_1| + ...