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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1answer
28 views

why a lemma shows well-definedness of linear transformations

The following lemma can be used to show that some linear transformations are well-defined. I don't quite see that. I mean, if a linear transformation $T$ is well-defined, then if $x=y$ then ...
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1answer
64 views

Banach space problem

I came across the following problem: For an open set $U$ in $\mathbb{R^n}$ we define the set of all k-times continuously differentiable functions $f:U\rightarrow \mathbb{R}$ for which $D^\alpha f$ ...
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1answer
118 views

Covering of closed unit ball with closed balls.

Notations and definitions Let $E$ be a finite dimensional vector space with norm $||\;||$. Let $B$ denote the closed unit ball in $E$ and $B_r[a]$ the closed ball centered at $a$ with radius $r$. ...
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2answers
78 views

What does this theorem mean?

Let $(V,\|\cdot\|)$ be a finite-dimensional normed space. Define $\|T\|_\mathrm{op}=\sup\{\|T(x)\|:\|x\|≦1\}$, for all linear operators on $V$ Define $\Omega$ to be the set of all invertible linear ...
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0answers
29 views

A System of Matrix Equations

I am looking for the solutions to the following system of matrix equations: (1) $AA^{*} - A^{*}A = C^{*}C - BB^{*}$ (2) $DD^{*} - D^{*}D = B^{*}B - CC^{*}$ (3) $AC^{*} - C^{*}D = A^{*}B - ...
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59 views

Largest/smallest cross-norm: A simple question about cross-norms on tensor products of Banach spaces.

This is a very simple dumb question as I’m completely new to the topic. I was reading Wikipedia’s entry on “Topological Tensor Products”, and there’s one thing I’m confused about. Let $ A $ and $ B $ ...
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0answers
134 views

The Hilbert space $\mathcal{H}_\eta$ and unitary correspondence with $L^2[a,b]$

The question I have is related to a problem in Stein and Shakarchi's Real Analysis, Chapter 4. The problem Let $\eta(t)$ be a fixed strictly positive continuous function $[a,b]$. Define ...
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1answer
55 views

Prove scalar product in a normed vector space is an open mapping.

I have the feeling this is a really obvious question, but I'm having trouble with it, here it goes: Let $(X, ||\,||)$ be a normed vector space over $K$, prove that $\odot:K\setminus\{0\}\times X ...
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1answer
42 views

What is *the domain* of the definition of the operator norm.

My definition: Let $(V,||\cdot||),(W,||\cdot||)$ be normed spaces over $\mathbb{F}$. Let $T:V\rightarrow W$ be a continuous linear transformation. Then $||T||_{op}\triangleq ...
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2answers
43 views

How do i prove that continuous linear transformation between normed spaces is bounded?

Let $(V,||\cdot||_V),(W,||\cdot||_W)$ be normed spaces over $\mathbb{C}$. Let $T:V\rightarrow W$ be a linear transformation. Assume $T$ is continuous. Then, how do i prove that $\exists c>0$ ...
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0answers
63 views

Understanding the right assumption about Gateaux derivative in Banach spaces

It is a general fact that the notion of Gateaux derivative is not uniform over the mathematical community i.e. someone requires it to be linear and continue other not and require this additional ...
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3answers
110 views

Showing that the norm of the canonical projection $X\to X/M$ is $1$

How do I show that given $M$ a closed subspace of a normed space $X$, and let $\pi$ be the canonical projection of X onto $X/M$. Prove that $\|\pi\| = 1$. I figure I could use Riesz' lemma and set ...
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2answers
68 views

a functional analysis question

$X$ is a banach space and $f$ a non zero linear functional. I'm trying to show $null(f)$ not dense in X $\implies f$ continuous. I've tried a few approaches but I think the following seems the most ...
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1answer
114 views

Duality mappings on finite-dimensional spaces

I have a few questions regarding some concepts in a book "nonlinear partial differential equations by Roubicek" I am studying. The following is from the text. "Let $V$ be a separable, reflexive ...
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1answer
58 views

Are $C^0[a,b]$ and $C^0[0,1]$ isometrically isomorphic?

Consider $C^0[a,b]$ and $C^0[0,1]$, each equipped with the $L^1$-Norm. Are these (out of curiosity) isometrically isomorphic?
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0answers
1k views

Natural matrix norm of an inverse matrix

Let $\left\|\cdot\right\| : \text{GL}(n,\mathbb{R})\to\mathbb{R}_{\ge 0}$ denote the natural matrix norm, i.e. $$\left\|A\right\|:=\max_{x\ne ...
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1answer
138 views

Inequality for norm of linear combination of linearly independent vectors

I'm trying to find a proof for the following: Let {$u_{1},...,u_{n}$} be a linearly independent set of a normed space $X$. Then, there is a constant $c>0$ such that for every set of scalars ...
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2answers
134 views

How does one prove $\left\lVert\sum\limits_{n=1}^{\infty}f_n\right\rVert\le \sum\limits_{n=1}^{\infty}\lVert f_n\rVert$?

Can the triangle inequality for norms works for infinite some as well? Hope that someone can give some hints to prove it or give a counterexample. Thanks!
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1answer
42 views

Cardinality of maximal subsets with some property

Let $X$ be a normed space. Is it true that all maximal (with respect to "$\subset$") subsets $D\subset X$ with the following property: $$ \|x-y\| \geq1 \textrm{ for } x\neq y, x,y\in D, $$ are of the ...
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1answer
195 views

Normed Vectors Spaces

Let $(E,\| \cdot \|_E)$ and $(F,\| \cdot \|_F)$ be two normed vector spaces over $\mathbb{C}$ and let $u: E\rightarrow F$ be a linear map. (a). Prove that the following conditions are equivalent: i. ...
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1answer
48 views

Clarifying the PDE notation C^1([0,T], X).

In studying nonlinear hyperbolic PDE, I've come across the following spaces: $C([0,T],H^s(\mathbb{R}^n))$. $C^1([0,T], H^s(\mathbb{R}^n))$. $L^p([0,T],H^s(\mathbb{R}^n))$. I presume that $(1)$ ...
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1answer
117 views

Are $L_p$ spaces of functions with separable support separable?

Let $X$ be a separable space. Is $L_p$$(X, \mu, V)$ a separable space? Here, $(V, |\cdot|_V)$ is a normed space. And a norm of $L_p(X, \mu, V)$ is: $$ \|f\|=\left(\int_X \big(\vert f\rvert_V\big)^p ...
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2answers
83 views

Can anyone explain this isometry to me? $T: (\mathbb R^n,\|\cdot\|_1)^*\to (\mathbb R^n,\|\cdot\|_{\infty})$, $ T(x)(y) = \sum_{i=1}^n x_i y_i$

Can anyone explain this isometry to me? $$T: (\mathbb R^n,\|\cdot\|_1)^*\to (\mathbb R^n,\|\cdot\|_{\infty}),\qquad T(x)(y) = \sum_{i=1}^n x_i y_i$$ I don't get what the domain and image of $T$ are. ...
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1answer
90 views

Hausdorff metric and convex hull

Let $X$ be a normed linear space and $A, B \in P(X)$. We define $\overline{co}(A) =$ the closure of the convex hull of $A$. Let $h$ denote the usual Hausdorff metric. We need to show that: $$h( ...
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0answers
127 views

Show that the normed space $(l^1, ||.||_1)$ is complete.

I am thinking to start off by saying that $\{x_n\}$ is Cauchy in $l_1$, so for every $\epsilon>0$, there exists an $N$ such that $\sum^\infty_{k=1}\mid x^n_k - x^m_k\mid <\epsilon^2$ for $n$, ...
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1answer
39 views

Convergence of a sequence in polynomials vector space

Let $N\in\mathbb{N}$ be a natural number and let $\alpha_0,\alpha_1,...,\alpha_N$ be real numbers such that $\alpha_i\neq\alpha_j \forall i\neq j$ We define in $X$ (the vector space of polynomials ...
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1answer
41 views

Inequality on Hilbert spaces in order to prove the nonexpansivity of a mapping.

I have an application $T\colon H\to H$ (where $H$ is a Hilbert space) such that $$(Tx-Ty,x-y)\leq \|x-y\|^2,\forall x,y\in H$$ where $(\cdot,\cdot)$ is the inner product of $H$ and $\|\cdot\|$ its ...
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1answer
72 views

question about norms and convex set

Suppose $\overline{B}(0;1) = \{ x \in X : ||x|| \leq 1 \}$ is the closed unit ball on a vector space $X$. MY question is: is the following true? If $\overline{B}(0,1) $ is not convex, then $|| \cdot ...
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3answers
67 views

Norm of vectors inequality

I tried proving this with triangular inequality but i was not right can any one help me with this
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2answers
158 views

Density of $C^{1}_{0}(\mathbb R)$ in $L^{\infty}(\mathbb R)$

I am looking for a counterexample to $C^{1}_{0}(\mathbb R)$ ( $C^1$ functions with compact support) is dense in $L^{\infty}(\mathbb R)$? Is there some easy counterexample showing that this latter is ...
2
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1answer
42 views

Norm of vector with respect to operator

Define $L$ is a linear operator maps from $E^n$ to $\mathbb{R}$, its norm is defined as $||L||_{op}=\sup\limits_{||x||=1}L(x)$, where $||\cdot||$ is any norm on $E^n$. How to show that ...
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5answers
231 views

The distance between two sets inside euclidean space

Let $A,B \subseteq \mathbb{R}^d$ be non-empty sets. Define their distance to be $$ d(A,B) = \inf \{ \|x-y\| : x \in A, \; \; y \in B \} $$ For any $A,B$, I want to prove that $d(A,B) = d( ...
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0answers
111 views

Difference between unconditional and absolute convergence in Banach spaces

One can show that in any finite-dimensional normed vector space absolute convergence is equivalent to unconditional convergence. It's not hard to show that if we have an orthonormal sequence in ...
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1answer
180 views

The dual space of normed vector space $X$ is isomorphic to the dual of its completion

Let $\overline{X}$ be Banach space and $X$ be dense subset of it. Show that dual space of X and dual space of $\overline{X}$ are isomorphic. Why these are isomorphic? I don't know how to prove ...
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1answer
48 views

Compact embedding

Prove that the embedding $j\colon (C^1[0,1],\|\cdot\|)\to(L^1[0,1],\|\cdot\|_{L^1})$ where $\|f\|=\max\{\|f\|_\infty,\|f'\|_\infty\}$ and $\|f\|_\infty$ denotes the supremum norm, ...
3
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1answer
449 views

Equality in Young's inequality for convolution

I am trying to understand how sharp Young's inequality for convolution is. The inequality says $||f \ast g||_r \leq ||f||_p ||g||_q$ where as $1/p+1/q = 1+1/r$. Actually, there are a couple of papers ...
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2answers
46 views

For inner product spaces, do we have $||\vec{u}-\vec{v}|| \leq ||\vec{u}||+||\vec{v}||$?

Let $V$ be an inner product space. Then for all $\vec{u},\vec{v} \in V$ we have $$||\vec{u}-\vec{v}|| \leq ||\vec{u}||+||\vec{v}||.$$ I know that the converse to the equation is true such that ...
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1answer
42 views

In a normed space, is it always true that $\|a_1e_1+\dots+a_ne_n\|\geq |a_i|\|e_i\|$?

In a normed space, is it true in general that $\|a_1e_1+\dots+a_ne_n\|\geq |a_i|\|e_i\|$ for all $1\leq i\leq n$? $e_i$ are basis elements of the vector. This is definitely true for the Euclidean ...
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1answer
45 views

Continuous linear functional

I want to show that $f:(\ell^1,\parallel. \parallel_1)\to \mathbb K$ defined by $f((x_n))=\sum\limits_{n=1}^{\infty}\dfrac{\vert x_n\vert}{n}$ is continuous linear functional and the norm of $f$ is ...
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3answers
124 views

Is any norm induced by some inner product? [duplicate]

It is a well-know fact that an inner product induces some norm. How about the converse? I think it's false but I can't think of an example. I'm thinking some properties like the parallelogram law ...
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1answer
23 views

Verifying whether a given function can be a norm.

I was asked to prove that given the vector space $\Bbb{R}\times\Bbb{R}$, the function $f(p)=(\sqrt{a}+\sqrt{b})^2$, where $p=(a,b)$, does not define a norm (on $\Bbb{R}\times\Bbb{R}$). Is the ...
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2answers
393 views

Gateaux and Frechet derivatives and related notions

Let $X$ and $Y$ be normed real vector spaces, and $f : X \to Y$ a map. Let's say that: G) $f$ is Gateaux differentiable at $x_0 \in X$ if for all directions $v \in X$ the limit $f'(x_0)(v) := ...
2
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1answer
86 views

On the completeness of inner product spaces.

Let $H$ be a Hilbert space, equipped with an inner product $(\cdot,\cdot)_1$ and norm $\|\cdot\|_1$ induced by it. Let $(\cdot,\cdot)_2$ be other inner product on $H$ and $\|\cdot\|_2$ the norm ...
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2answers
103 views

TVS: Topology vs. Scalar Product

I just had an idea: It is clear that every scalar product induces a norm and that a metric and that finally a topology. Turning this argument around we know: Not every topology induces a metric only ...
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1answer
94 views

Is $H^2\cap H^1_0$ dense in $H_0^1$?

Let $-\infty<a<b<+\infty$ and $I=(a,b)$. Consider $H^1_0(I)$ equipped with the norm $\|\cdot\|_{H^1}$ given by $$\|f\|_{H^1}=\|f\|_{L^2}+\|f'\|_{L^2}.$$ Is $H^2(I)\cap H_0^1(I)$ dense in ...
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1answer
21 views

Range of a continuous linear mapping

I want to show that the range of the linear map $T:(\ell^1,\parallel .\parallel_1)\to (\ell^2,\parallel .\parallel_2)$ defined by $Tx=x$ is not closed. I considered a sequence $(x^{(n)})$ in ...
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2answers
89 views

Inequality regarding norm vector space

I am not sure how to prove this inequality involving norms. Let $X$ be a normed vector space and $x,y$ are vectors in $X$ with nonzero norms. Prove the following inequality is true. $$\|x-y\|\geq ...
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1answer
23 views

What happens to $l_1$ if i change coordinate system.

Let $x =(x_1,\ldots,x_n) \in \mathbb{R}^n$ and also $x=\sum_{i=1}^m t_i u_i,$ where $t_i \in \mathbb{R}$ and $u_i \in \mathbb{R}^n.$ Is it true that $||x||_1 \geq \sum_{i=1}^m |t_i|$ ?
3
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1answer
201 views

Weak* continuity

Let $B$ be the open unit ball in $\mathbb{R}^2$ and $\mathcal{M}^+$ the set of nonnegative Radon measures on $B$ and $\mathcal{M}^2$ the set of $\mathbb{R}^2 \text{-valued}$ Radon measures on $B.$ I ...
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0answers
59 views

Weak convergence, weak neighborhoods

Let $V$ be a normed vector space, $V'$ its continuous dual. Let $U \subset V$. Consider the statements: i) For any finite $F \subset V'$ there exists $y \in U$ with $\max_{f \in F} |f(y)| < 1$. ...