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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$L^p$ is a quasi normed space for $0<p<1$ [duplicate]

I know that $L^p$ is a vector space for $p>0$ and a normed space for $p \geqslant 1$ now I need show that for $ 0<p<1$ and $f,g \in L^p$ exist $K \in \mathbb{R}$ such that $||f+g||_p ...
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49 views

Hermite functions as eigenvectors of Fourier transform

In order to find an orthogonal basis of eigenvectors of the Fourier transform operator $F:L_2(\mathbb{R})\to L_2(\mathbb{R})$, $f\mapsto\lim_{N\to\infty}\int_{[-N,N]}f(x)e^{-i\lambda x}d\mu_x$ for ...
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1answer
12 views

Weighted Norm Minimization

I have a minimization problem of the form $min (w_1 \|x\|+w_2\|y\|)$ subject to constraints $A_1x=b_1$, $A_2y=b_2$, $0< l_1 \leq x \leq u_1$ and $0< l_2 \leq y \leq u_2$ where ...
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12 views

Composition of two functions in normed spaces

Let $\Omega_1, \Omega_2 \subset \mathbb{R^n}$ be bounded. The mapping $ F: \Omega_1 \rightarrow \Omega_2 $ shall be bijective, continuously differentiable and such that $||DF(x)||$ and ...
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50 views

Erwine Kryszeg's _Introductory Functional Analysis With Applications_: Section 2.3, Prob. 14

Here's problem 14 in the Problem Set immediately following Section 2.3 in the book, Introductory Functional Analysis With Applications by Erwine Kryszeg. Let $Y$ be a closed subspace of a normed ...
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1answer
81 views

Are all the finite dimensional vector spaces with a metric isometric to $\mathbb R^n$

Are all the finite dimensional vector spaces with a metric isometric to $\mathbb R^n$? My goal is to claim that in any finite dimensional vector space, equipped with a metric, a closed-bounded subset ...
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1answer
23 views

Some Norms on $V=P_n(\mathbb{R})$?

$V=P_n(\mathbb{R})$ be the vector space of all polynomials with degree $\le n$. I need to know Which of the followings are norms on $V.$ $\forall p\in V$ 1.$\|p\|^2=|p(1)|^2+\dots+|p(n+1)|^2$ ...
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39 views

Continuity of vector space operations in a normed space

Here's problem 4 immediately following section 2.3 in Erwine Kryszeg's book, Introductory Functional Analysis With Applications: Show that in a normed space $X$, vector addition and scalar ...
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1answer
107 views

Prove that the subset $X$ of a normed vector space $(V,\|\cdot\|)$ is complete.

My subset $X$ has the Bolzano-Weierstrass property and I need to prove that $X$ is complete in the sense that every Cauchy sequence in $X$ converges to a point in $X$. I know that having the ...
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43 views

Theorem 2.3-2 in _Introductory Functional Analysis With Applications_ by Erwine Kryszeg

Here's the statement of Theorem 2.3-2 in the book mentioned above: Let $(X,||\cdot||)$ be a normed space. Then there is a Banach space $\hat{X}$ and an isometry $A \colon X \to W$, where $W = A(X)$, ...
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40 views

examples on double dual spaces

I am looking for examples on double dual spaces as I know $\ell_p $ is a double dual of $\ell_p$ for $1<p,q<\infty$. $\mathcal L_p $ is a double dual for $\mathcal L_p$ for ...
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1answer
22 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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1answer
46 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
31 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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1answer
32 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
19 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
52 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
17 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
26 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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27 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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23 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
52 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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83 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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25 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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20 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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27 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
27 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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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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1answer
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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1answer
11 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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30 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
43 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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308 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
56 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
31 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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19 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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24 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
23 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
27 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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37 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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51 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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32 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
37 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
44 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 ...