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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200 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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50 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
78 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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76 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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33 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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33 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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34 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
87 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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97 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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45 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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38 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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2answers
133 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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44 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
40 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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158 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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42 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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24 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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58 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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55 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
42 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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28 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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414 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
31 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
325 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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84 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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27 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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41 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
36 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
34 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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69 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
38 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
46 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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100 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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2answers
50 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
31 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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66 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
49 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
165 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
165 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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82 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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43 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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48 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| + ...
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1answer
53 views

Unit Ball in 1 norm is open in ($C[0,1] , || \quad ||_{\infty}$)

Claim $B_1(0,1) := \{ f \in C[0,1] ; ||f||_{1} < 1 \} $ in $(C[0,1],||\quad || _1)$ is open in $(C[0,1],||\quad || _{\infty}).$ We need to take any $f \in B_1(0,1),$ and we have to find an ...
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147 views

Completely metrizable space, complete norm

Let $E$ be a normed, completely metrizable space. Prove that the initial norm is complete. How can I go about solving this problem? I will be grateful for all your hints. Thank you!
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2answers
520 views

Norm equivalence (Frobenius and infinity)

I am trying to prove the matrix norm equivalence for norms 1, 2, $\infty$ and Frobenius. I have managed to solve find the constants for $||.||_{1}$ and $||.||_{2}$ but I cannot see how to continue ...
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1answer
59 views

How to compute the induced matrix norm $\| \cdot \|_{2,\infty}$

The induced norm of the matrix $A$ as a map from $(\mathbb R^n , \| \cdot \|_p)$ to $(\mathbb R^n, \| \cdot \|_q)$ is given by $$ \| A \|_{p,q} = \sup_{x\in\mathbb{R}^n\setminus \{0\}} ...
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1answer
162 views

Lipschitz continuity and gradient of a real-valued function on a normed space

The gradient of a function $f:E\to\mathbb{R}$ is Lipschitz continuous with parameter $L > 0$ iff $$\|\nabla f(x) - \nabla f(y)\|^* \le L\|x-y\| \quad \forall x,y\in E.$$ I have two questions: ...
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1answer
57 views

Show that a subspace of a normed vector space is closed

Let $X$ be a normed vector space over $\mathbb K, \mathbb K = \mathbb R$ or $\mathbb K=\mathbb C.$ Let $Y$ be a closed linear subspace of $X$ and $x\in X\backslash Y.$ Set $Z=\{y+\alpha x;\;y\in ...
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1answer
60 views

Prove that a subset of an infinite dimensional complete space is uncountable

Let $X$ be a Banach space. A subset $S ⊂ X$ is called a Hamel basis of $X$ if $S$ is linearly independent and every element of X is a finite linear combination of elements of $S$. (i) Prove that if ...
3
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1answer
107 views

Computing the norm of a linear operator

For two finite-dimensional real vector spaces $E_1,E_2$, define an linear operator $A:E_1\to E_2^*$. Its adjoint operator is defined by $A^*:E_2\to E_1^*$ its adjoint operator, i.e. $$\langle Ax,u ...