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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43 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
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 ...
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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 ...
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3answers
74 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
21 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
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2answers
186 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) := ...
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1answer
60 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
64 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 ...
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1answer
60 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
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 ...
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2answers
64 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
22 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
110 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
51 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
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2answers
52 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
165 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
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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 $f$ ...
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1answer
76 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
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0answers
63 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
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1answer
155 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 ...
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1answer
80 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, ...
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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 ...
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3answers
216 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 ...
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1answer
66 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 ...
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0answers
50 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 ...
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1answer
31 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 ...
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2answers
145 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 ...
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1answer
226 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 , ...
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1answer
74 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 ...
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1answer
116 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, ...
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0answers
86 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
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1answer
64 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 ...
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0answers
85 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 ...
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2answers
79 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?
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3answers
103 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 ...
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1answer
146 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 ...
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1answer
328 views

Norm of Matrix transpose

I have a problem below: Let $\|\cdot\|$ denotes the norm matrix \begin{equation} \|A\|=\max \frac {\|Ax\|}{\|x\|}, \end{equation} for every $A$. Now suppose that $H: \mathbb{R}^k \rightarrow ...
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1answer
129 views

Closure of $B=\{ f\in C'[0,1] : |f(x)|\leq 1, |f'(t)|\leq 1 \forall t\in[0,1]\}$ (NBHM $2005$)

If $A$ is the closure in $\mathcal{C}[0,1]$ of the set $B$ where $$B=\{ f\in C'[0,1] : |f(x)|\leq 1, |f'(t)|\leq 1 \forall t\in[0,1]\}$$ Then Which of the following is true? $A$ is closed. $A$ ...
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1answer
27 views

Bounded & Norm space [closed]

Can someone help me on this exercise ? Thanks!
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1answer
172 views

Isomorphisms between Normed Spaces

Between two normed (linear) spaces there are several notions of isomorphisms: Linear isomorphisms: linear bijective maps Topological isomorphisms: linear homeomorphisms (due to the linearity these ...
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1answer
51 views

About the closedness of Banach space

I'm reading a proof in trying to prove that a Banach space $X$ is reflexive if and only if $X^{*}$ is reflexive. There's a point in the proof saying that $X$ is a closed subspace of $X^{**}$, but I ...
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0answers
85 views

Linear Operators: Continuous $\Rightarrow$ Bounded

Let $T:V\rightarrow V'$ be a continuous linear operator between two normed vector spaces $V,V'$. Show that it is bounded. Continuity is defined as $\lim_{n}\|x_n-x\|=0\Rightarrow ...
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2answers
131 views

Are the two infima equal?

Let $R$ be a ring (not necessarily commutative) or an algebra over some field with a norm defined on it and let $I$ be an ideal in $R$. Let $a,b \in R$. Does it hold that $\inf_{i,j \in I}\|ab + ai ...
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3answers
108 views

If every inner product space can be converted into a norm space, then why is there a distinction between the two?

If every inner product space can be converted into a norm space, then why is there a distinction between the two? $$\|x\| = \sqrt{\langle x,x\rangle }$$
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0answers
87 views

normed space functional analysis [closed]

applying this lemma for e1=(1,0,0) e2=(0,1,0) e3=(0,0,1) what is maximum of c? i found c=1 am i right?
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2answers
102 views

Completeness/Compactness of a subset in a normed linear space

Let $(X,\|\cdot\|)$ be the normed linear space consisting of the sequences $a=(a_n)_{n=1}^{\infty}$, for which the corresponding series $\sum_{n=1}^{\infty} a_n$ converges absolutely, with norm ...
2
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1answer
56 views

Endpoint-average inequality for a line segment in a normed space

Let $X$ be a normed vector space over $\mathbb R$. What is the smallest universal constant $C>0$ such that the inequality $$\|x\|\le C\int_0^1 \|x+tv\|\,dt\tag{1}$$ holds for all $x,v\in X$? ...
2
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1answer
67 views

The proof of the triangle inequality of this norm

Let $V$ be a normed vector space and $W$ a closed subspace of $V$. It is possible to define a norm on $V/W$ (the quotient space) by defining $\|v + W \|_{V/W} = \inf_{w \in W} \|v + w\|$. As an ...
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1answer
34 views

Proof of uniqueness of zero for this norm

Let $V$ be a normed vector space and $W$ a closed subspace of $V$. It is possible to define a norm on $V/W$ (the quotient space) by defining $\|v + W \|_{V/W} = \inf_{w \in W} \|v + w\|$. As an ...
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0answers
34 views

Proof about scalar multiplication of this norm

Let $V$ be a normed vector space and $W$ a closed subspace of $V$. It is possible to define a norm on $V/W$ (the quotient space) by defining $\|v + W \| = \inf_{w \in W} \|v + w\|$. As an exercise I ...