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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2answers
38 views

norm of canonical projection = 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 ...
1
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2answers
56 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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0answers
25 views

Is the set, $\{f\in Y: \left\|f\right\|_{Y} \leq C \ \text{and} \ \left\|(|f|)\right\|_{X}\leq C \}$, closed in $(Y, ||\cdot||_{Y})$?

Put, $X= \text{The space of "nice" complex valued functions on } \mathbb R; \text{that is}, f:\mathbb R \to \mathbb C;$ so that $X$ is Banach space with respect to the norm ...
5
votes
1answer
63 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
50 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
77 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 ...
1
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1answer
45 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 ...
2
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2answers
108 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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0answers
52 views

Triangle Inequality for matrix norm

$A$ and $B$ are square matrices, show that $σ_{max}(B) ≥ |σ_{min} (A+B) - σ_{min} (A)|$ I know that the induced 2-norm of of $B$ is $σ_{max}(B)$ but I don't know how to proceed with RHS.
2
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1answer
40 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 ...
3
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1answer
188 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
34 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 $(1)$ ...
3
votes
1answer
79 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 ...
2
votes
2answers
77 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. ...
1
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1answer
38 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( ...
0
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0answers
71 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 $n$, ...
0
votes
1answer
31 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 ...
0
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1answer
35 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
52 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 ...
0
votes
3answers
55 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
116 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
votes
1answer
30 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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0answers
33 views

Prove line segments are always geodesics in a normed vector space

The length of a path $\gamma: [0,1] \rightarrow \mathbb{V}$ in some normed vector space $\mathbb{V}$ is defined by $$l(\gamma):=sup \sum_{i=1}^n ||\gamma(t_i)-\gamma(t_{i-1})||$$ where the supremum ...
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0answers
52 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
76 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 ...
0
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0answers
25 views

subjectivity of transpose and bounded from below

This is problem of Tao's epsilon of the room 1.5.13. Let $T : X \to Y $ be a continuous linear transformation which is bounded from below (i.e. there exists $c > 0$ such that $\|Tx\| \geq ...
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1answer
33 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
119 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
40 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 ...
-1
votes
1answer
39 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 ...
0
votes
1answer
37 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 ...
0
votes
3answers
66 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 ...
1
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1answer
20 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 ...
3
votes
2answers
154 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
votes
1answer
56 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 ...
2
votes
2answers
61 views

Topology induced by 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 ...
0
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1answer
55 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 ...
0
votes
1answer
18 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 ...
1
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2answers
61 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 ...
0
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1answer
21 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
votes
1answer
100 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 ...
1
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0answers
46 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 $\max_{f \in F} |f(y)| < 1$. ...
3
votes
2answers
48 views

Why is the image of a compact operator separable?

Let $A$ and $B$ be normed vector spaces and let $S\in \mathscr{K}(A,B)$ be a compact operator. Question: How does it follow that the image of $S$ is separable? Thanks for the help.
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4answers
157 views

Why is $C_c^\infty(\Omega)$ not a normed space?

I am watching a Coursera video on Théorie des Distributions and I am trying to understand one of the slides. Let $\Omega \subset \mathbb{R}^N$ be an open set and $C_K^\infty(\Omega) = \{ \phi \in ...
2
votes
1answer
48 views

Show that E\H (Hyperplane) is arc-connected $\Longleftrightarrow$ H isn't a closed subspace

Good morning, Let $E$ be a real normed vector space and $H$ a hyperplane of $E$ Show that E\H is arc-connected $\Longleftrightarrow$ H isn't a closed subspace I have no idea to solve it. But If $f$ ...
0
votes
1answer
71 views

Problem in harmonic analysis

suppose $p$ be a fixed psitive real number and $f$ is an entire function with $$\lvert f(0) \rvert^p=\int_\mathbb{C}\ \lvert f(z)\exp(-\alpha\lvert z \rvert ^2) \rvert^p dA(z) $$ where $\alpha ...
2
votes
0answers
62 views

Space of Continuous mappings to metric spaces

I want to ask whether some basic result from the space $C([0,1],R)$, where $R$ is the real space carries over to the space $C([0,1],E)$, where $(E,\|\cdot\|_E)$ is a metric space. We know that ...
2
votes
1answer
141 views

If the dual spaces are isometrically isomorphic are the spaces isomorphic?

Let $X$, $Y$ be Banach spaces such that the duals $X^\ast$ and $Y^\ast$ are isometrically isomorphic. Are $X$ and $Y$ necessarily isomorphic? The answer to the question whether $X$ and $Y$ are ...
2
votes
1answer
73 views

The density of diagonalizable matrices of $M_n(\mathbb{C})$ problem.

For any matrix $A = (a_{ij})_{1\leq i,j\leq n} \in M_n(\mathbb{C})$, we pose $||A|| = \max_{1\leq i,j\leq n} |a_{ij}|$. $1.$ Show that $||.||$ define a norm on $M_n(\mathbb{C})$ and that $\forall A, ...
0
votes
1answer
40 views

How to detect reflexivity of the closure

Consider the space of continuous bounded functions on a bounded interval. Its closure for the Lebesgue $L_p$ norm is reflexive when $1 < p < \infty$, but it is not reflexive for $p = 1$. How ...