1
vote
0answers
35 views

Operator norm of the inverse

If I made no mistake, one can calculate the operator norm of the inverse of any given (invertible) operator $A: V\rightarrow V$ via: \begin{align}\|A^{-1}\| & = ...
0
votes
0answers
17 views

Dose the closed unit ball of C(the closure of E) with sub-norm, have no extreme points?

Let E be a bounded closed set in R^n. Dose the closed unit ball of C(E) with sup-norm, have no extreme points?
1
vote
2answers
20 views

How to prove this inequality for operator and function

How to prove this? $\sum_{k=1}^{\infty}|(Tf)_k|^2\leq ||T||^2||f||^2$ where $T$ is an operator and a function $f$. $(Tf)_k$ is the $k$-th coordinate of Tf. Should this involve Cauchy-Schwarz or the ...
1
vote
2answers
65 views

Is is true that $||v+w||^2 = ||v||^2 + 2\langle v,w \rangle + ||w||^2$?

Is it true that for a $\mathbb{R}$ vector space with dot product $\langle\cdot, \cdot\rangle$ and $||\cdot||$ norm \begin{align} ||v+w||^2 = ||v||^2 + 2\langle v,w\rangle + ||w||^2 && ...
0
votes
0answers
22 views

Show that $\lim_{a \to \infty} \sup_{n} \int_0^{T-a}||v_{n,r}(t+a)-v_{n,r}(t)||_{\mathbb{L}^2(\Omega_{2r})}^2dt=0.$

Let $\ 0 \leq t \leq t+a \leq T$, with $$\lim_{a \to 0} \sup_{n} \int_0^{T-a}\left\|u_n(t+a)-u_n(t)\right\|_{\mathbb{L}^2(\Omega_{r})}^2dt=0,$$ where $\Omega_r=\Omega \cap \left\{x \in \mathbb{R}^2; ...
0
votes
0answers
21 views

Uniformly Bounded in a Schauder Norm

Suppose $\{h_{k}\}$ is a sequence of elements in $C^{2,\alpha}$. I want to show that $\{h_{k}\}$ is uniformly bounded in $C^{2,\alpha}$. Now, if I'm not mistaken, then $||h_{k}||_{\displaystyle ...
1
vote
1answer
22 views

Check continuity and find norm of linear functional

I have to check continuity and find norm of the following linear functional $$\ell^{2}\ni \{x_n\}^{\infty}_{n=0} \rightarrow \sum^\infty_{n=0}\frac{x_n}{\sqrt{2^n}}\in \mathbb{K}$$ As for continuity, ...
0
votes
1answer
29 views

Normed linear space with two norms that are not equivalent, one is complete, what about the other?

I have been searching for an answer to the following question: Given a normed linear space $V$ and two norms that are not equivalent, but $\exists K\in\mathbf{R}$ such that $\|v\|_1\leq K\|v\|_2$ ...
1
vote
1answer
33 views

Sobolev Inequality with powers

The following is one of the Sobolev Inequalities as stated in Gilbarg & Trudinger's text on elliptic PDEs: $\displaystyle W_{0}^{1,p}(\Omega)\subset L^{\frac{np}{(n-p)}}(\Omega)$ for $p<n$. ...
0
votes
1answer
31 views

What is the usual norm of $\mathbb{H_0^1}$?

What is the usual norm of product of sobolev spaces $\mathbb{H_0^1}=H_0^1 \times H_0^1=W^{1,2}\times W^{1,2}$? In my work i need to prove that the norm endowed by the inner product ...
1
vote
1answer
31 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 ...
1
vote
1answer
22 views

Norm of linear functional

For every $ x \in C([a,b])$, we define the functional $F(x) = \sum_{i=1}^{n} {\lambda_{i} x(t_{i})}$ where $\lambda_{i} \in R, i=1,...,n$. I was wondering if someone can help me to find a sequence ...
2
votes
2answers
49 views

Comparable norms on the space of polynomials?

Are the norms: $$\|P\|_1=\int^1_0\|P(t) \| dt\mbox{ and }\|P\|_2=\sup_{0\le t\le1} |P(t)|.t$$ comparable on the vector space $X=\mathcal{P}[0,1]$, i.e. all polynomials on $[0,1]$? Here, i try to ...
0
votes
0answers
24 views

Why this relation in Hilbert space (with inner product $< >$) holds?

$c_k$, $f_k$ are sequences in Hilbert space, $g$ is a function. Why this relation below holds? How you derive it? $\sum_k|c_k\left \langle f_k,g \right \rangle|\leq(\sum_k|c_k|^2)^{1/2}(\sum_k|\left ...
0
votes
2answers
37 views

Why $||f||=\mathrm{sup}_{||g||=1}|\langle{f,g}\rangle|$

My question is why $||f||=\mathrm{sup}_{||g||=1}|\langle{f,g}\rangle|$? for any two functions $f$ and $g$ with $||g||=1$, and $||\;||$ denotes the 2-norm. I have tried to use the triangle inequality ...
0
votes
0answers
31 views

Conditions on $\alpha_n$ for $n_\alpha(x)=\sum^\infty_{n=1} \alpha_n\vert x_n\vert $ to be a norm on $l_p$

When $n_\alpha(x)=\sum^\infty_{n=1} \alpha_n\vert x_n\vert $ is a norm in $\mathcal{l}_p=\lbrace (x_k)^\infty_1 : \sum\vert x_k\vert ^p \lt\infty\rbrace $ and $\alpha\in\omega$. and $\omega$:space of ...
1
vote
1answer
37 views

Proof check for completeness

I'd like to see if the proof I have is adequate. Statement. Let $X$ and $Y$ be Banach space, the product $X\times Y$ is a vector space under coordinate operations with norm $$ \|(x,y)\| = \|x\|_X ...
1
vote
1answer
47 views

When sup norm,i.e. $\| x|| = \sup|x(t)|$ for $\forall t\in T $in C[0,1] for $T \subsetneq [0,1]$

When sup norm, i.e. $\| x|| =\sup|x(t)|$ for $\forall t\in T $in C[0,1] $T$ is such that $T \subsetneq [0,1]$. What condition should be applied to $T$ to make $\| x||$ a norm. I cannot show for ...
-1
votes
1answer
31 views

bounded subset of normed space

Suppose $X$ is a normed linear space and $S\subset X$. Show that if $$\sup_{x\in S}\{\mid f(x)\mid\}<\infty$$ for every $f\in X^{\ast}$, then $S$ is a bounded in norm.
0
votes
1answer
23 views

Example for a two norms and vector space which are equivalent on this Vectorspace

I need an example where the two norms and the vector space are equivalent on the vector space.
0
votes
3answers
53 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
1answer
42 views

$l^p$ is super subset of $ l^q$ if and only if?

$$ l^p \subseteq l^q. $$ if and only if ? $l^p$ is a subset of $l^q$ when is it possible ? !
2
votes
1answer
28 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 ...
5
votes
0answers
82 views

renorm a Banach space to make an operator have spectral radius equal to norm

Let $X$ be an infinite-dimensional complex Banach space equipped with the norm $\lVert\cdot\rVert$, and let $T\in\mathcal{L}(X)$ a bounded linear operator on $X$. Let $r(T)$ denote the spectral ...
2
votes
1answer
39 views

Weak Lower Semicontinuity Generalized to any $L_{p}$ space

I am having difficulty with the following proof: Generalize the weak lower semi-continuity of$L^{p}$ norms to all $1\leq p < \infty$; i.e., show that if $u_{n}\to u$ weakly in $L^{p}$, then ...
4
votes
0answers
51 views

norm of differential operator on $P^n[0,1]$

Consider the space $P^n[0,1]$ of polynomials of degree $\leq n$ on $[0,1]$, equipped with the sup norm. Now, this is a finite dimensional space, so all linear operators have to be continuous, hence ...
1
vote
0answers
52 views

What is the correct notation for defining norms in measure spaces?

For a function $f \in L_2(R)$, we can define its norm as $$ \|f\|_2^2 = \int f^2(x) dx $$ If I use a different measure $\mu$, I can in turn define the norm as $$ \|f\|_2^2 = \int f^2(x) \mu (x) dx ...
2
votes
1answer
57 views

Any positive linear functional $\phi$ on $\ell^\infty$ is a bounded linear operator and has $\|\phi \| = \phi((1,1,…)) $

This is a small exercise that I just can't seem to figure out. When I see it I'll probably go 'ahhh!', but so far I haven't made any progress. I'd like to prove that any linear functional $\phi$ on ...
2
votes
1answer
33 views

Showing that a map defined using the dual is a bounded linear operator from X' into X'

I have trouble answering the second part of the following exercise. Any help would be appreciated! Let $(X, \| \cdot \|)$ be a reflexive Banach space. Let $\{ T_n \}_{n = 1}^\infty$ be a sequence of ...
2
votes
2answers
45 views

how to prove that the limit of this sequence of functions is continuous?

I have a norm that works in function space and that is $‖∙‖_{sup}:C([0,1])→R$, $‖∙‖_{sup}:=sup${$|f(t)|$}. I need to show that the metric is complete. So I need to show that every Cauchy sequence of ...
1
vote
1answer
69 views

addition and multiplication of functions in function space, continuous?

I have a norm that works in function space of C[0,1]. How do I show that addition and multiplication of functions (C[0,1]xC[0,1]->C[0,1]) are continuous functions?
0
votes
1answer
51 views

Check that: it is a norm

My question is: Let $f \in C^{1} [0,1]$ & let $f'$ denote its derivative. Define: $|| f ||_{1} = ( \int_{0}^{1} (|f(t)|^{2} + |f'(t)|^{2})dt)^{\frac{1}{2}}.$We are to show that: $||f||_{1}$ ...
0
votes
1answer
49 views

A matrix in $SL(2)$ has it's supremum norm and infimum fulfilled by orthogonal vectors

I am having trouble proving the next statement: If $B\in SL(2)$ and $||B||\neq 1$, for $||B||:= \underset{x\neq 0}{\sup}\big\{\frac{||B(x)||}{||x||} \big\} $, where $||\cdot||$ is the euclidian norm, ...
1
vote
1answer
96 views

Why is the weak* topology not in general metrizable?

A Banach space is a topological group under addition. The dual is a topological group under the weak$^*$ topology. The weak$^*$ topology is weaker than the operator norm topology, so is it ...
0
votes
1answer
87 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 ...
0
votes
2answers
73 views

Prove $\|f\|_{L^p}$ is not equivalent to $\|f\|_{\infty}$ in $C[a,b]$

Prove that in $C[a,b]$ the uniform norm is not equivalent to the $L^p$ norm for $(1\leq p < \infty)$ I am stuck on showing that the function below satifies the claim. I know that f is continuous ...
1
vote
1answer
28 views

The space $\mathcal{D}((0,T);V)$ and its norm/embeddings?

Let $V$ be a Hilbert space. Define $\mathcal{D}((0,T);V)$ to be the set of functions $u:(0,T) \to V$ such $u$ is compactly supported on $(0,T)$ and is a $C^\infty$ test function. What is the norm ...
2
votes
1answer
42 views

Representation in Banach space and norms 'induced' by representation

By $G$ we denote some compact group, $X$ stands for some Banach space. Suppose $\pi\colon G\longrightarrow \mathrm{GL}(X)$ to be some representation in $X$. I'm trying to prove that there is an ...
2
votes
1answer
35 views

Derivative of function that includes norm

I was solving the problem: find the derivative of a function f : H → R, $f (x) = \sin ||x||^3$ (H is Hilbert space). I got the answer $f'(x)=3\cos||x||^3 x||x||$. Is this correct or I am doing ...
0
votes
2answers
234 views

Show these two norms are not equivalent?

I have the following two norms on $C[a,b]$ : $$||x||_1= \int_a^b |x(t)|dt$$ $$||x||_\infty = sup_{t \in [a,b]}|x(t)|$$ $\forall x \in C[a,b]$. I need to prove that these are not equivalent. They are ...
1
vote
1answer
42 views

Prove this sequence in $L(\ell_1)$ does not converge to zero?

I have a linear operator $$A: \ell_1 \rightarrow \ell_1$$ $$A_nx=(0,0,...,0,x_{n+1},x_{n+2},...)$$ with $n$ zeros. I am asked to show two things: that for any $x \in \ell_1$, $\lim_{n \rightarrow ...
0
votes
3answers
68 views

Compute the norm of matrix

Let $M$ be $n\times n$ matrix, consisting entirely of 1's. Show, that $\|M\|_{op}=\sup_{x\in C^n}|Mx|=n$.
1
vote
1answer
48 views

Relation of norms of matrices

Let $A$ be $m \times n$ matrix. Let $B=\frac 1n AA^*$, where $A^*$ is a transposed matrix. Let $X_i, I\leq m$ be row-vectors of $A$. Show $$ \|B\|=\frac 1n \|A\|^2\geq \max_{i\leq m}|X_i|, $$ Where, ...
1
vote
1answer
44 views

continuity of $L^p$ norms with respect to $p$

Let $0<p_0<p<p<p_1\leq \infty$. Then I have proved $L^{p_0}(\mu)\cap L^{p_1}(\mu)\subseteq L^{p}(\mu)$. In particular, when $p_0=1$, $p_1=\infty$, I have proved further ...
1
vote
1answer
42 views

Norm of operator $T_x(f) = f(x)$

Let $X$ be a normed vectorspace and $X'$ be the dual space of $X$. For $x \in X$ we can define $T_x: X' \to \mathbb F$ by $T_x(f) := f(x)$. This is indeed an operator in $X''$. I read that $\| T_x \| ...
1
vote
2answers
64 views

Prove that $d_{1}$ and $d_{2}$ induce the same topology

Suppose $d_{1}(x,y) = |x-y|$, $d_{2}(x, y) = |\phi(x) - \phi(y)|$, where $\phi(x) = \frac{x}{1 + |x|}$. Prove that $d_{1}$ and $d_{2}$ are metrics on $\mathbb{R}$ which induce the same topology. ...
2
votes
1answer
73 views

There exists function sequence $\{f_{n}\}$ converges to $0$ such that $\{a_{n}f_{n}\}$ not converges to $0$

Let $X$ be the vector space of all complex functions on the unit interval $[0,1]$, topologized by the family of seminorms $$p_{x}(f) = |f(x)|, \quad (0 \le x \le 1).$$ Show that there exists a ...
2
votes
1answer
49 views

Triangle inequality in product space of normed spaces

Let $(X,||.||_X)$ and $(Y,||.||_Y)$ be normed spaces, then $||(x,y)||:=(||x||_X^p+||y||_Y^p)^{\frac{1}{p}}$ is a norm on $X \times Y$. This is absolutely clear to me, but I have troubles to verify ...
1
vote
1answer
44 views

Can we write $||A - B|| \leq ||A||$?

I am confused with the very basic question related with the matrix norm. Can we write $||A - B|| \leq ||A||$ ? Thanks for the help and time.
0
votes
1answer
54 views

Prove that $|x(a)| + \max \{|x'(t)|: t \in [a,b]\}$ is a norm for a complete space

Prove that the space $C^1([a,b])$ consisting of continuous functions in $[a, b]$ with the norm $|x(a)| + \max \{|x'(t)|: t \in [a,b]\}$ is a Banach space. I can't prove the completeness of this ...