Tagged Questions
1
vote
1answer
27 views
Lipschitz condition normed vector space
Am I right that $g: C^{1}[0,2],||*||_{C^1[0,2]} \rightarrow \mathbb{R}$ with $g(f)=f'(1)$ satisfies a Lipschitz condition?
Cause $|f'(1)-h'(1)|=|g(f)-g(h)|\le \text{max} |f-h|+ \text{max} |f'-h'|$, ...
2
votes
1answer
44 views
Norm of an operator
How to find the norm of the following operator
$$
A:\ell_p\to\ell_p:(x_n)\mapsto\left(n^{-1}\sum\limits_{k=1}^n k^{-1/2} x_k\right)
$$
Any help is welcome.
8
votes
2answers
126 views
Basic question about $\sup_{x\neq 0}{} \frac{\|Ax\|}{\|x\|} = \sup_{\|x\| = 1}{\|Ax\|} $, $x \in\mathbb{R}^n$
I am having trouble with understanding the following definition while studying some basic things related with matrix norms:
For every matrix $A\in M_n(\mathbb{R})$
$$\sup_{x\neq 0}{} ...
2
votes
1answer
66 views
Inequality for norms
Let g(x, y) be function on $X\times Y$. Show that for all $p\geq q$
$$
\|\,\|g\|_{L^q(Y)}\,\|_{L^p(X)}\leq \|\,\|g\|_{L^p(X)}\,\|_{L^q(Y)}
$$
Thsnk you.
1
vote
1answer
29 views
Simple norm equivalence
Please I need help with this problem.
Let $V$ be the vector space consisting of functions which are continuous over the interval $[0,1]$, take the value $0$ at the points $0$ and $1$ and are once ...
3
votes
2answers
115 views
Can the $0$-norm represent determinism?
In Scott Aaronson's Quantum Computing since Democritus, he presents classical probability theory as based on the $1$-norm, and QM as based on the $2$-norm.
Call $\{v_1,\ldots,v_N\}$ a unit vector ...
4
votes
1answer
70 views
Proving an alternative norm on Sobolev space is equivalent to usual norm
I have this exercice and my problel is only in item 4, and i will desespere.
Let $f \in L^2(\mathbb{R}^n).$
1- Why the equation $\Delta u - u = \dfrac{\partial f}{\partial x_i}$ admits a unique ...
3
votes
3answers
58 views
Example of two norms on same space, non-equivalent, with one dominating the other
I am looking for an example (with proof) of two norms defined on the same vector space, such that the norms on the two spaces are NOT equivalent, but such that one norm dominates the other...
4
votes
1answer
98 views
Linear isometry between $c_0$ and $c$
The following question is an exercise and so I'm just looking for advices and not for answers if it's possible.
I have the following sets in $l^\infty$
$$c_0 := \{x_n \in l^\infty: \lim x_n = 0\} ...
1
vote
1answer
22 views
product of bounded linear operators
If I have 2 bounded linear operators $T_1,T_2$ such that $T_2:X\rightarrow Y$ and $T_1:Y\rightarrow Z$. I know that by boundedness, $||T_2(x)||\leq||T_2||\,||x||$ and using the norm of $T$ defined as ...
0
votes
1answer
47 views
$V$ is finite dimensional iff $V'$ with the weak topology is normable
Why is the following statement valid? Note, $V$ is locally convex Hausdorff topological vector space over $\mathbb{C}$ and $V'$ is the space of all continuous linear maps from $V \to \mathbb{C}$.
...
3
votes
0answers
65 views
Prove this equality in a Hilbert Space $H$ with Riesz's Representation Theorem.
For $x \in H$, prove that $\sup_{\|z\| = 1} \langle x , z \rangle= \| x \|$
Here is a quick one. If someone could improve this it would be great
Proof
By Cauchy Schwarz, $\langle x,z \rangle ...
1
vote
2answers
39 views
Linear functional $\mathscr{L}(E,F)$
Let $\mathscr{L}(E,F)$ denote the space of all linear functionals from $E \to F$.
Let $\mathscr{C}(E,F)$ denote the space of continuous linear functionals from $E \to F$. My question:
How to prove ...
1
vote
1answer
43 views
Study the equivalence of these norms
I have two Hilbert spaces $H_1$ and $H_2$ and I consider a set of functions $f$ which decompose as $f=g+h$ with $g\in H_1$ and $h\in H_2$. I know that this decomposition is unique.
So I define the ...
0
votes
1answer
28 views
What is $L^1$ norm for $n$-fold functional space of integrable function?
Consider a space of $n$-fold real-valued integrable functions, $X \doteq [L^1([0,T])]^n$. Some use an alternative notation like $X \doteq L^1([0,T]; \mathbb{R}^n)$.
Does $f\in X$ mean ...
0
votes
0answers
28 views
All matrix/vector norms induce the same topology?
From Wikipedia
all norms on $K^{m \times n}$ are equivalent; they induce the same topology on $K^{m \times n}$. This is true because the vector space $K^{m \times n}$ has the finite dimension $m ...
1
vote
1answer
62 views
Calculating the Norm of an operator in $L^2(0,1)$
If I have the following operator for $H=L^2(0,1)$:
$$Tf(s)=\int_0^1 (5s^2t^2+2)(f(t))dt$$ and I wish to calculate $||T||$, how do I go about doing this:
I know that in $L^2(0,1)$ we have that ...
1
vote
2answers
57 views
Are the norms $\|\cdot\|_1$ and $\|\cdot\|$ equivalent?
Let $\|x\|_1=\sum_{n=1}^{\infty}|x_n|$ is a norm for $\ell^1$, suppose $x=\{x_n\}\in\ell^1$ and $\|x\|=\sup|\sum_{k=1}^{n}x_k|$.
Are the norms $\|\cdot\|_1$ and $\|\cdot\|$ equivalent?
1
vote
0answers
22 views
A modular which is not a metrizing modular (hence not an F-norm)?
I'm taking the terminology from Rolewicz's 1985 Metric Linear Spaces.
Given a complex vector space $X$, a modular $m$ is any function $m:X\to[0,+\infty]$ satisfying the following for all $x,y\in X$ ...
4
votes
1answer
132 views
Using Lax Milgram to find a weak solution in an intersection of Sobolev spaces
I am trying to prove the existence of a weak solution of the problem:
$$
-\Delta^2 u = f \in L^2(U)\\ \\
u|_{\partial U}=\Delta u|_{\partial U} = 0
$$
on the bounded open set $U\subset\mathbb{R}^n$ ...
2
votes
1answer
29 views
Simple doubt about dual norm
If $(X, \|\cdot\|)$ is a normed vector space, then
$$\|F\|_{X^{\prime}}\ =\ \sup_{x\in X-\{0\}}\frac{|F(x)|}{\|x\|},$$
by definition. Then I want prove that,
$$\|F\|_{X^{\prime}}\ =\ ...
2
votes
1answer
32 views
Regularity and the Varitational Inequality
Let $K = \left\{ v \in H_0^1(\Omega) \, : \, v \geq 0 \right\}$, further suppose $\Omega$ has the regularity property that $||v||_{H^2} \leq C(\Omega)||\Delta v||_{L^2}$, for all $v \in ...
3
votes
1answer
82 views
Norm of the linear functional
Could you help me, please with the following question?
There is a linear functional $A : C_{[0;1]} \rightarrow \mathbb{R}$, such that
$$ Ax=\int_{a}^{b}x(t)\varphi(t)dt $$
where $\varphi$ is a fixed ...
0
votes
1answer
35 views
Norm of a function, Smoothness Penalization
I am seeking for some intuition why norm (for any reasonable norm on functions) of a function is smaller if the function is smoother.
2
votes
3answers
73 views
norm of a linear operator
On http://www.proofwiki.org/wiki/Definition:Norm_(Linear_Transformation) , it is stated that $||Ah|| \leq ||A||||h||$ where $A$ is an operator.
Is this a theorem of some sort? If so, how can it be ...
1
vote
1answer
13 views
Positivity of a map in $(l^\infty(X))^*$
Let $X$ be a set and $\varphi: l^\infty(X)\to\mathbb{R}$ be a linear map such that
$||\varphi||=1$
$\varphi(1_X)=1$
I am trying to prove that $\varphi(f)\ge 0$ for all $f\ge 0$, but all my ...
0
votes
2answers
42 views
Norm on vector space.
I was given a true or false questionnaire to study for my final and do not know if I am right or wrong about these statements. I marked the following statement as True:
If $\|\cdot\|$ is a norm on ...
2
votes
1answer
55 views
finding operator norm $T_N$
How do find the operator norm of , $T_N\colon c_0\to \Bbb R$ given by
$T_N(y):=\sum_{j=1}^Nx_jy_j$ ,when $N$ is a integer .
TIA
2
votes
1answer
81 views
Invertible operator norm bound
Let $H$ be a Hilbert space and that $X$ are bounded. Suppose
$X$ is self-adjoint. Show that $Y=X+iI$ is invertible and the inverse $Y^{-1}$ has the norm $\lVert Y^{-1} \rVert \le 1$.
I can prove $Y$ ...
6
votes
2answers
86 views
Counterexamples of Arzèla Ascoli theorem for non-obeyed criteria
I had an exam on functional analysis some time ago, and one of the questions I couldn't make any sense out was the following:
Let $\Omega\subset \mathbb{R}$ and $\{f_n\}$ a sequence of continuous ...
2
votes
1answer
51 views
A specific linear operator between Banach spaces
Let B be the Banach space $B=(C[0,1],\|\cdot\|_{\infty}$) and let $\{\xi_i\}\in l^\infty$.
Let $T:l^1\rightarrow B$ be the linear operator given by: $(Ta)(x) = \sum_n\xi_na_nx^n$.
I have three ...
1
vote
2answers
60 views
Operator norm converging to 0 for certain condition
Let $X$ be a finite-dimensional normed space and $T_n : X \to X$ a sequence of linear operators such that $\lim_nT_nx = 0$ for all $x$ in $X$. Prove that $\lim_n\|T_n\|=0$.
3
votes
0answers
74 views
The norm of an operator
Let $\rho(x)$ be a weight function in a unit sphere, such that
\begin{equation}
\begin{array}{l}
\displaystyle 1. \rho(x)\ge 0,\int_{\mathbb{R}^n}\rho(x)=1\\
\displaystyle 2. \rho(x)\in ...
3
votes
2answers
94 views
Two proofs concerning Hölder's inequality
I am studying functional analysis and I have come across two statements which can be proven by using Hölder's inequality, but I don't know how/why. Reminder: let $a\in l^p$ and $b\in l^q$, then:
$$
...
0
votes
1answer
129 views
The $ l^{\infty} $-norm is equal to the limit of the $ l^{p} $-norms.
If we are in a sequence space, then the $ l^{p} $-norm of the sequence $ \mathbf{x} = (x_{i})_{i \in \mathbb{N}} $ is $ \displaystyle \left( \sum_{i=1}^{\infty} |x_{i}|^{p} \right)^{1/p} $.
The $ ...
6
votes
2answers
115 views
equivalence of norms
I would like a little help here:
I have two defined norms over $C^{1}([0,1])$ :
$\| A(f)\|=|f(0)|+\max_{x\in[0,1]}{|f'(x)|}$
$\| B(f)\|=\int_0^1|f(x)|dx+\max_{x\in[0,1]}{|f'(x)|}$
I already ...
3
votes
1answer
67 views
Operator norm and spectrum
Let $L$ be a bounded linear operator on a Hilbert space. Without assuming finite dimensions, can we express the operator norm of $L$ in terms of the spectrum of the positive operator $L^{\dagger}L$?
...
2
votes
1answer
99 views
Norm equivalence Sobolev space
I have this problem:
Let $k>0$ (integer) and $1 \leq p < \infty$. Show that the norms
$$ ||u||_{W^{k,p}(U)} = \bigg( \sum_{|\alpha|\leq k}||D^{\alpha}u||_{L^{p}(U)}^{p}\bigg)^{\frac{1}{p}} $$
...
0
votes
1answer
70 views
Sobolev spaces doubt
Can somebody help me with this doubt?
Let $\Omega$ an open set and $A$ be any finite subset of points of $\Omega.$
Is it true the following inequality?
$\vert v(a) \vert \leq C \| v \|_p ...
0
votes
3answers
162 views
How to find the norm of this bounded linear functional?
Let $C^\prime[a,b]$ denote the normed space of all continuously differentiable real (or complex) valued functions defined on the closed, bounded interval $[a,b]$ in $\mathbf{R}$ with the norm defined ...
2
votes
2answers
84 views
Norm of differentiation operator $Tf(t)=f^{'}$..
Consider $T:C^1[0, 1]\rightarrow C[0, 1]$ given by $Tf=f'$ where $$\|f\|_{C^1}=\|f\|_\infty+\|f'\|_\infty$$ and $\|f\|_\infty=\sup_{x\in [0, 1]}|f(x)|$. How to prove $\|T\|=1$? The inequality ...
9
votes
1answer
224 views
On the norm of a quotient of a Banach space.
Let $E$ be a Banach space and $F$ a closed subspace. It is well known that the quotient space $E/F$ is also a Banach space with respect to the norm
$$
\left\Vert x+F\right\Vert_{E/F}=\inf\{\left\Vert ...
0
votes
1answer
23 views
A limit superior question in the context of the Neumann series
I'm trying to understand a step in the proof that the Neumann series converges:
Let $X$ be a Banach space and $T\in L(X)$ (the space of bounded, i.e. continuous linear maps $X\to X$). It is known ...
1
vote
1answer
74 views
Matrix Trace representation?
For a real, symmetric matrix $A$ and a real, rectangular matrix $X$, am looking for a matrix trace based representation of this simple linear algebraic expression $\sum_{i} A_{ii} ...
0
votes
1answer
47 views
Norm of normal Operator A
I just found the following equality $||A||=sup_{\lambda\in\sigma(A)} |\lambda|$
My question: What exactly means $\sigma(A)$ and why this is true ?
I always thouht the only way to get the ...
0
votes
1answer
69 views
Hilbert norm and Euclidean distance
For real matrix $X$ where $d_{i,j}^2(X)$ indicates the euclidean distance squared between the rows $i,j$ of $X$, if $d_{i,j}^2(X)=||f(X_i.)-f(X_j.)||_H$ then what would the function $f(.)$ be? Is ...
2
votes
2answers
95 views
equivalence of norms and direct sum
Let $(X,\|\cdot\|_X) $ be an infinite dimensional Banach space. Suppose that you can write $X=V\oplus W$. Write $x=v+w$ and define $(V\oplus W,\|x\|_\oplus :=\|v\|_X+\|w\|_X)$. Show that ...
1
vote
1answer
77 views
Equivalence of Norms Defined on a Cartesian Product
While studying some notes on normed vector spaces, I have come upon the proof that addition $+:V \times V\to V$ of vectors in a normed vector space $V$ is a continuous operation.
The proof of this ...
1
vote
1answer
115 views
Dual norm and distance
Let $Z$ be a subspace of a normed linear space $X$ and $x\in X$ has distance $d=\inf\{||z-y||:z\in Z\}$ to $Z$.
I would like to find a function $f\in X^*$ that satifies
$||f||\le1$, $f(x)=d$ and ...
3
votes
0answers
69 views
Absolute norms and 1-unconditional sums
Absolute norm
Let $X$ and $Y$ be Banach spaces. Let $Z=X\times Y$ a norm $\|\cdot\|_N$ on $Z$ is called absolute if there is a function $N\colon R^2\rightarrow R$ such that
$$
\|(x,y)\|_N=N((\|x\|, ...
