4
votes
1answer
93 views

On an estimate of sequences with weights

Does there exist a $C > 0$ such that $$ \sum_{n \geq 1} a_n \leq C \left( \sum_{n \geq 1} 2^n a_n^2 \right)^{1/4} \left( \sum_{n \geq 1} 2^{-n} a_n^2 \right)^{1/4} $$ for all $a_n \geq 0$ with ...
0
votes
1answer
36 views

Value of $\frac1{\Vert f\Vert}$

For $V=\{x\in X\mid f(x)=1\}$ show that $\inf\{\Vert x\Vert\mid x\in V\}=\frac1{\Vert f\Vert}$, where $X$ is banach and $f$ is a nontrivial element of the dual space of $X$. For $x\in V$ we have ...
1
vote
1answer
46 views

Space of Distribution wrt to topology of uniformly convergence on bounded sets not Frechet-Space.

I found a state, that the Space of Distribution on (here:) $\mathbb{R}^{n}$, which is equipped with the topology of uniformly convergence on bounded subsets is not a Frechet Space. As far as i can ...
1
vote
1answer
37 views

Show that a set is compact.

Let $X$ be a Banach space and $\{A_t\}_{t\in R}$ a family of linear and continuous maps $X \rightarrow X$ such that function $\mathbb{R} \ni t\rightarrow \|A_t x\| \in \mathbb{R}$ is continuous for ...
1
vote
1answer
38 views

Find the norm of functional

Consider the functional from $l_2$. $$ x=(x_n)\mapsto \sum \frac{x_n+x_{n+1}}{2^n}. $$ What is the norm of the functional?
0
votes
1answer
32 views

How can you compare two mathematical functions of single variable? [closed]

I have some functions and want to see how close one function is to the other. One way to check is correlation. What are the other ways of checking the "similarness", "closeness" or "nearness"?
1
vote
0answers
45 views

Find norm of linear operators

I have to check if those operators are bounded and if so what are their norms. 1) $\phi:C^1[0,1]\ni f > \rightarrow\int_0^{1/2}f(t)dt+f'(2/3)\in\mathbb{R}$ with norm ...
1
vote
1answer
41 views

Inequality with a norm

I need help with the following: Let $A=\left(\begin{array}{cc}a & b \\c & d\end{array}\right)$, with $a\in\mathbb{R}$, $b\in(l^{1})^{*}$, $c\in l^{1}$, and $d\in L(l^{1},l^{1})$. Let $h\in ...
1
vote
1answer
71 views

Show that $\limsup_{n\rightarrow\infty} \|x_n\|^{\frac{1}{n}}=1$

Let $(x_n)$ be weakly convergent, but not norm convergent, sequence from a Banach space. Show that $\limsup_{n\rightarrow\infty} \|x_n\|^{\frac{1}{n}}=1$. Any help?
0
votes
1answer
20 views

help about find the norm in$L^1$ [duplicate]

I need to find norm of $f$.Firstly I try to show boundedness and so I need to find any $M>0$.But coefficient of integral is zero.Where is my mistake? and I used holder inequality for norm in ...
1
vote
1answer
24 views

Norm of the functional $f(x)=\int_0^2 x(t).(t^2-1)dt$ in the space $X=L_1(0,2)$.

I am trying to calculate the norm of the functional $f(x)=\int_0^2 x(t).(t^2-1)dt$ in the space $X=L_1(0,2)$. What we have is: $||f(x)||\le ||x||_1.||t^2-1||_{\infty}$ by extended Hölder's ...
0
votes
0answers
33 views

Question about functions in Sobolev space.

Let $\Omega$ be a bounded Lipschitz domain in $\mathbb{R}^n$. If I consider a function $g:\mathbb{R}\rightarrow\mathbb{R}$ which has the following properties: $$ |g(x)|\leq M \qquad |g(x)-g(y)|\leq ...
1
vote
1answer
105 views

Fredholm operator norm

I have seen here, that the operator norm of a Fredholm operator $T_k(f)(s):=\int_0^1 k(s,t) f(t) dt $, where $k \in L^2([0,1]^2)$ and $f \in L^2([0,1])$ is not equal to the $L^2$ norm of the Kernel. ...
0
votes
1answer
31 views

Geometric representation of norm of an operator [closed]

Let $T$ is a bounded linear transformation from a normed linear space $X$ to normed linear space $Y$. What is the Geometric representation of $||T||$? Hint: $$\|T\|=\sup\{k:||Tx||\leq k||X||\}$$
5
votes
1answer
62 views

Missing a necessary power in this proof - please help.

This question is somewhat related to Gradient Estimate - Question about Inequality vs. Equality sign in one part. That question was related to part (c) of a problem I am working on, and this question ...
0
votes
1answer
32 views

Hilbert Schmidt norm inequality

I was wondering if anyone knows about an inequality for the Hilbert-Schmidt (H-S) norm of the type $|Tr (Bg)|\leq Const\cdot||B||\cdot function(||g||_{2})$ for a bounded operator $B$ and a H-S ...
1
vote
1answer
47 views

Norm of an operator defined on sequence spaces

Consider the sequence space $\ell_r$ defined by $$\ell_r=\left\{x=(x_n)_{n=1}^{\infty}:x_n\in\mathbb{R}\text{ and }\sum_{n=1}^{\infty}|x_n|^r<\infty\right\}.$$ Let $2\leq p,q<\infty$ such that ...
1
vote
1answer
53 views

Are the following norms equivalent?

We have the norms $||f||_1=||f||_\infty+||f'||_\infty$ and $||f||_2=|f(a)|+||f'||_\infty$ where $f\in C^1[a,b]$. Are they equivalent and how shoud I prove/disprove this.
0
votes
1answer
34 views

an $L^p$ implication

Let $1<p<\infty$ and $x,y\in L^p([0,1])$ such that $\|x\|_p = \|y\|_p = 1$. Then the following implication holds: $$\left\|\frac{x+y}{2}\right\|_p=1\Rightarrow x=y\tag{*}$$ This ...
0
votes
1answer
27 views

norm on a quotient-space

Let $M:[0,\infty)\to[0,\infty)$ be continuous and convex. Further $M$ satisfies $M(t)=0\Leftrightarrow t=0$. Let $$\mathcal L_M(\mathbb R):=\left\{f:\mathbb R\to\mathbb R \mathrm{\ measurable\ ...
1
vote
1answer
25 views

Check continuity of linear functionals and find norms

1) $c_{00} \owns (x_n) \mapsto \sum_{n=0}^{\infty} x_n \in \mathbb{K}$ where $c_{00}$ is a space of sequences that are eventually equal to $0$ with sup norm 2) $\ell^\infty \owns (x_n) \mapsto ...
1
vote
0answers
88 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
1answer
34 views

Does the closed unit ball of $C(E)$ have no extreme points?

Let $E$ be a bounded closed set in $\mathbb R^n$. Does the closed unit ball of $C(E)$ (the space of continuous functions on $E$ with supremum norm) have no extreme points?
1
vote
2answers
23 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 ...
0
votes
2answers
69 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
23 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; ...
1
vote
0answers
29 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
37 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
73 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
41 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 $p<n$. ...
1
vote
1answer
41 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
34 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
31 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
64 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
25 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
32 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
38 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
50 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 ...
0
votes
1answer
28 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
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
0
votes
1answer
46 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
29 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
106 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
43 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
87 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
53 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
75 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
35 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
47 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 ...