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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2
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2answers
82 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
vote
1answer
81 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( ...
1
vote
0answers
125 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
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 ...
0
votes
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 ...
1
vote
1answer
71 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
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
0
votes
2answers
155 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
41 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 ...
2
votes
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( ...
1
vote
0answers
96 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 ...
0
votes
1answer
167 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 ...
1
vote
1answer
46 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
votes
1answer
427 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 ...
1
vote
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 ...
-1
votes
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 ...
0
votes
1answer
44 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
113 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
vote
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 ...
4
votes
2answers
361 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
79 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
102 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 ...
0
votes
1answer
88 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
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 ...
1
vote
2answers
85 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
votes
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
votes
1answer
186 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
vote
0answers
58 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
94 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.
2
votes
4answers
212 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
61 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
83 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
74 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
259 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
142 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
44 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 ...
2
votes
3answers
491 views

Sum of closed subspaces of normed linear space

Problem Suppose $R$ is a normed linear space, then show that: If $M$ is closed subspace of $R$ and $N$ a finite dimensional subspace of $R$, then the set $$M+N=\{ z : z = x + y , x \in M , y \in N ...
1
vote
1answer
96 views

Preannihilator of the image of an adjoint of a bounded operator

Let $E,F$ be normed spaces and $F\colon E\rightarrow F$ be a linear bounded operator. Denote by $$A'\colon F'\rightarrow E'$$ the adjoint of the operator between the topological duals of the normed ...
1
vote
1answer
75 views

Complete inner product space always has an orthonormal basis

I am trying to understand inner product space. My text book says: A complete linear space with scalar product(not necessarily seperable) always has an orthonormal basis. I looked up the ...
0
votes
1answer
33 views

Check whether a sequence belongs to an open ball

How to check if the sequence x=( x1 , x2 ,...) where xn =1-(1/n) belongs to the open ball B(0,1) in the normed space l^∞ of all bounded sequences with the norm defined by ...
4
votes
2answers
420 views

Collecting things that are preserved by (isometric) isomorphisms between normed spaces

I would like to collect a list of things that are preserved under isomorphisms and isometric isomorphisms. The reason is that I hope to get a better perception of their importance in Functional ...
0
votes
1answer
385 views

Unit balls in normed spaces.

Assume we talk about the $n$ dimensional vector space over the reals. It is easy to see that for any norm the unit ball is a convex symmetric set. And here is my question : Let $A$ be a bounded , ...
2
votes
1answer
95 views

Convergence in Sobolev Spaces

Consider the bounded mapping $A:W^{1,p}(\Omega) \rightarrow W^{1,p}(\Omega)^{*}$ where $A$ is defined as: $\langle A(u),v \rangle\text{ } := \int_{\Omega}a(x,u,\nabla u)\cdot \nabla v + c(x,u,\nabla ...
4
votes
1answer
253 views

Euclidean space without orthonormal basis

I've been thinking about: Problem Give an example of a nonseperable Euclidean space which has no orthonormal basis. My Argument I know if a Euclidean space $R$ has at most countable basis, ...
-1
votes
0answers
104 views

Absolutely convergent series in normed linear space

I want to prove that in a normed linear space $X$ if for all absolutely convergent series $\sum\limits^{\infty}_{n=1}x_n$, the series $\sum\limits^{\infty}_{n=1}T(x_n)$ is convergent, then $T:X\to Y$ ...
2
votes
1answer
87 views

Algebraic dimension of infinite-dimensional Banach Space

I am trying to show: Algebraic dimension of infinite-dimensional Banach Space is uncountable. By algebraic dimension it is meant that the cardinality of the Hamel Basis of the space. Suppose ...
1
vote
0answers
98 views

Every inner product space is a normed space which is also a metric space

As I am pretty sure that everybody knows that a Hilbert space is a space that is a complete, separable and generally infinite dimensional inner product space. By the means of completion, every Cauchy ...
2
votes
2answers
87 views

restriction a non compact operator to compact operator

If $T\in\mathcal{B}(X,Y)$ is not compact can the restriction of $T$ to an infinite dimensional subspace of $X$ be compact?
1
vote
3answers
107 views

Is there a norm on ${\Bbb R}^{\Bbb N}$

Let $E={\Bbb R}^{\Bbb N}$ be the real vector space of real sequences. 1) Is there a norm on $E$? 2) Is there a norm $N$ on $E$ such that the restriction of $N$ to $\ell^2$ is finer than the ...
3
votes
1answer
256 views

Weak continuity in Sobolev Spaces

First consider the following two Sobolev Embedding Theorems. Theorem 1: The continuous embedding $W^{1,p}(\Omega) \subset L^{p^{*}}(\Omega)$ holds provided the exponent $p^{*}$ is defined as ...