Norm is a function on a vector space $X$ which generalizes notion of length of vector in general vector spaces.
2
votes
1answer
95 views
The Principle of Condensation of Singularities
Let $X$, $Y$ be Banach spaces and $\{T_{jk} : j,k \in\Bbb N\}$ be bounded linear maps from $X$ to $Y$. Suppose that for each $k$ there exists $x\in X$ such that $\sup\{\lVert T_{jk} x\rVert : j ...
2
votes
1answer
55 views
Distance between point and linear Space
Suppose $E$ is a normed vector space. Let $f$ be a continuous linear functional on $E$ and denote by $M$ the Kernel of $f$. Let $x\in E$. How to show that ...
0
votes
1answer
79 views
convergence in $L^2$
Let $ (f_n)$ be a sequence of measurable functions on $ [0,1]$ such that $\int_{0}^1 |f_n|^2 dm \le C$.
Assume that there exist a function $f$ such that $\int_{0}^1 |f_n-f|dm \to 0$ as $n \to \infty ...
1
vote
1answer
243 views
why is frobenius norm of a matrix greater than or equal to the 2 norm?
How can you prove that:
$$\|A\|_2 \le \|A\|_F$$
I cannot use:
$$\|A\|_2^2 = \lambda_{max}(A^TA)$$
It makes sense that the 2-norm would be less than or equal to the frobenius norm but I dont know how ...
2
votes
0answers
149 views
Equivalence of norms is a equivalence relation
Two norms $||-||_1 $, $||-||_2$are equivalent if:
for two constants $a,b$ and $x$ from $V$ a vector space over a field it holds that : $$a||x||_1\leqslant ||x||_2\leqslant b||x||_1.$$
This is a ...
2
votes
1answer
167 views
Does the limit of a convergent sequence depend on the norm?
Let $X$ be a vector space, and $\|\cdot\|_1$ and $\|\cdot\|_2$ two different (non-equivalent) Norms on $X.$ Let $(x_n)\subset X$ be a sequence and $x\in X$ such that $\lim_{n\to\infty}\|x_n-x\|_1=0.$ ...
2
votes
3answers
103 views
Proving a matrix equality
I have 2 matrices: $A \in R^{nxn}$ is a non-singular matrix and $B \in R^{nxn}$ is a singular matrix. Here is the expression I need to prove:
$$||A - B|| \ge ||A^{-1}||^{-1}$$
I dont understand why ...
1
vote
2answers
321 views
prove matrix norm equivalence
Given $A \in R^{m\times n}$, I need to prove:
$$||A||_2 \le \sqrt {m}||A||_\infty$$
I have tried a number of things and I just cant seem to get it to work.
Also, I need to prove:
$$||A||_2 \le ...
2
votes
1answer
117 views
Inequality of Lebesgue integral with $L^p$-norm
Let $X_t(\omega)$ be a continuous function $t\rightarrow L^p(\omega)$ (i.e., if we fixed the variable $t$ we obtain a function which belongs to $L^p$), with $t\in[0,T]$ and $\omega\in\mathbb{R}$.
I ...
2
votes
1answer
251 views
How to prove these three norm equivalence problems
Given $A \in R^{m\times n}$, I have these three norm equivalence equations:
$\|A\|_2 \le \|A\|_F \le \sqrt {n}\|A\|_2$
$\frac {1} {\sqrt n}\|A\|_{\infty} \le \|A\|_2 \le \sqrt {m} \|A\|_{\infty}$
...
0
votes
0answers
120 views
How to find the gradient and Hessian of the following function?
I have the following function $f(x)=\|x-a\|^2+\sum_{i=0}^{n-1}\sqrt{(x_{i+1}-x_i)^2+b}$ where $a$ is a n-vector and $b$ scalar.
I need to find the gradient and the Hessian of $f(x)$.
I am guessing ...
1
vote
1answer
42 views
Does switching between different $L_p$ norms preserve order?
Suppose you have a list of $n$ dimensional vectors. One can order them by using an $L_p$ norm to do comparisons between vectors. The general questions is, will the order be different depending on the ...
1
vote
3answers
54 views
Two norms on $C_b([0,\infty])$
$C_b([0,\infty])$ is the space of all bounded, continuous functions.
Let $||f||_a=(\int_{0}^{\infty}e^{-ax}|f(x)|^2)^{\frac{1}{2}}$
First I want to prove that it is a norm on $C_b([0,\infty])$. The ...
4
votes
2answers
213 views
I am not sure how to calculate this norm?
I have the following matrix: $$A=
\begin{bmatrix}
1 & 0 & 0 \\
0 & 1 & 1 \\
0 & 0 & 1 \\
\end{bmatrix}
$$
What is the norm of $A$? I ...
0
votes
1answer
52 views
How to find if the vector is stable?
Let's say I have the following equation $Ax=b$
My question is - how can i find a vector $b$ around which the above equation is not stable?
I have $$A=
\begin{bmatrix}
1 & 0.999 \\
...
5
votes
1answer
85 views
relation between norms of two vectors
Can we say that;
if the $l_1$-norm of an arbitrary vector $a$ is smaller that $l_1$-norm of $b$ ($||a||_1 \le ||b||_1$) then the $l_2$-norm of $a$ is smaller than $l_2$-norm of $b$
($||a||_2 \le ...
1
vote
1answer
41 views
Determine operator norm of mutiplication operator
Consider
$$T: (C[-1,1],\|\cdot\|_{2})\rightarrow \mathbb{C}\\Tf :=\int_{-1}^{1}mf\,\mathrm{d}x$$ where $m\in C[-1,1]$.
I want to prove $\|T\| = \|m\|_2$.
$\|T\|\leq\|m\|_2$ can be easily proved by ...
0
votes
0answers
60 views
on norm of submatrix of the inverse and inverse of a submatrix
Given an M-matrix, say $M\in\mathbb{R}^{n\times n}$, which in block form is
$M$ = \begin{pmatrix}
A & B\\
C & D
\end{pmatrix}
where $A\in\mathbb{R}^{k\times k}$ and ...
2
votes
1answer
90 views
Equivalence of two norms
Define two norms as following: $$
\left\Vert f\right\Vert _{1}={ \max_{0\leq x\leq1}\left|f\left(x\right)\right|}
, \quad\text{ and }\quad \left\Vert f\right\Vert ...
0
votes
2answers
345 views
How to do the following projection in Matlab?
I have 2 vectors $u$ and $v$ given in $\mathbb R^4$, e.g. $u = (-1,-2,3,4)$ and $v=(1,-2,-3,5)$
I also have $Ax=b$ which is an under-determined system; meaning, if $A$ is $m\times n$, then $m\le n$.
...
2
votes
1answer
72 views
How can I give a bound on the $L^2$ norm of this function?
I came across this question in an old qualifying exam, but I am stumped on how to approach it:
For $f\in L^p((1,\infty), m)$ ($m$ is the Lebesgue measure), $2<p<4$, let
$$(Vf)(x) = ...
1
vote
1answer
495 views
showing that l2 norm is smaller than l1
How can I show that L2<=L1
$||x||_1\le \cdot ||x||_2$
and also
$\|x\|_2\leq \sqrt m\|x\|_{\infty}$
regarding the first part, can I say that:
$$
\sqrt{\sum\limits_{i=1}^n x^2 } \leq ...
2
votes
1answer
118 views
Need help with relative and absolute errors?
Lets assume I have $Ax=b$ equation, where $A$ is $2$x$2$ matrix.
1) I want to find an A, x, and b such that relative error in x is small but absolute error in x is large
2) Also want to find A, x, ...
0
votes
0answers
67 views
How can I simplify this function according to given function?
Can anyone help me to solve this problem, please? I have this function as
$$f(x)=\int_{-\infty}^{\infty}\frac{1}{\sqrt{2\pi}\sigma_{\theta}}\exp\left(\frac{
\theta ...
0
votes
0answers
85 views
Gradient norm and BFGS method
I am using the Broyden–Fletcher–Goldfarb–Shanno (BFGS) method to find the minimum of a multivariable function. In the software implementation of the method I am using the stopping condition is only on ...
1
vote
1answer
70 views
Estimating the integral in norm.
I want to estimate the integral $$\int k(x,y)f(y)dy$$assuming the fact that $k(x,y), f(y)$ are in $L^p, L^q$ respectively.
But I want to bound the the whole integral in $L^r$, $r\in [1,\infty]$.
I ...
1
vote
1answer
135 views
operator norm and spectral radius
is it true that the operator norm of a matrix $A$ is smaller than 1 if its spectral radius $\rho(A)$ is smaller than 1?
many thanks for any help, it is much appreciated!
1
vote
1answer
34 views
Norm of two operators
(1) $U(x)=a \langle x,b \rangle +b \langle x,a \rangle $, $a,b\in H\setminus \{0\}$. U is an operator from H to H and a,b are orthogonal elements. I want to calculate $||U||$
For this one I tried the ...
3
votes
1answer
132 views
Proof of an inequality of $L^p$ norms
For a general measure space, we define : $\|f\|_p= \left(\int\vert f\vert^p du\right)^{1/p}$.
Let $0 < a < b < c < \infty$ and prove the following:
$$
\|f\|_b \leqslant \max\{\|f\|_a, ...
1
vote
1answer
83 views
How to calculate norm of operator in Hilbert space
LEt $H_1,H_2$ are two Hilbert spaces. $\{e_1,\ldots,e_n\}\subseteq H_1$ and $\{f_1,\ldots,f_n\}\subseteq H_2$ two orthonormal systems. $\lambda_1,\ldots,\lambda_n\in\mathbb K$. Let
$$
U:H_1\to ...
2
votes
0answers
72 views
Confusion about matrix norms
Reading the wiki article I get confused about matrix norms. My question, is it true that
$$\lVert Mx \rVert \leq C(\sup_{ij}{M_{ij}}) \lVert x \rVert$$
where $M$ is a matrix and $x$ is a vector and ...
2
votes
1answer
91 views
Equivalence between $Lip \ norm$ and $C_1 \ norm$.
Let $f\in C^1([a,b])$. Prove that $\|f\|_{C^1} = \|f\|_{Lip}$.
By definition of Lip norm and $C^1$ norm, it is equivalent to prove that $\|f'\|_{\infty}=Lip(f,(a,b))$, where the second member is the ...
0
votes
1answer
61 views
What does RMSD mean?
Normally a rigid superposition which minimizes the RMSD is performed, and this minimum is returned. Given two sets of points and , the RMSD is defined as follows:
$$\begin{align*}
...
0
votes
1answer
43 views
Bounding L2 distance with mean and standard deviation
Let $\mathbf{x}=[x_i]_{i=1}^d, \mathbf{y}=[y_i]_{i=1}^d$ be two vectors in $R^d$.
Is it possible to find a lower bound $l\leq \|x-y\|$ and an upper bound $u\geq\|x-y\|$ as a function of ...
1
vote
1answer
49 views
How to show that the scalar product on a vector space extends by continuity to a scalar product on the completion of the vector space?
I'm trying to solve the following problem:
Assume $H_0$ is a vector space equipped with a scalar product. Complete $H_0$ with respect to the norm $\Vert x \Vert = \langle x,x \rangle^{1/2}$. We ...
2
votes
1answer
155 views
Prove unit ball of $L^p(\mu)$ is strictly convex, when $1<p<\infty$
When $1<p<\infty$, $||f||_p=1$,$||g||_p=1$,$f\ne g$,then $\frac{1}{2} ||f+g||_p<1$.
I use parallelogram law
$||f+g||^2+||f-g||^2=||f||^2+||g||^2=4\\$
Since $f\ne g$,
$||f-g||^2>0$
...
3
votes
1answer
104 views
Finding the norm of the operators
How do I find the norm of the following operator i.e. how to find $\lVert T_z\rVert$ and $\lVert l\rVert$?
1) Let $z\in \ell^\infty$ and $T_z\colon \ell^p\to\ell^p$ with $$(T_zx)(n)=z(n)\cdot ...
2
votes
1answer
314 views
proving $L^\infty$ norm inequality (disprove $\Vert f\Vert_\infty\le\sqrt{n}$)
There are three parts in this question, I've done the first two but not sure about the third one. Also see $L^2$ norm inequality.
In the third part, I am asked to show that if $W$ is a ...
1
vote
2answers
106 views
Find the norm of an operator on $\ell_2$
Let $(x_n) \subset \ell_2$ and let operator $L:\ell_2\to \mathbb R$ be defined by:
$\displaystyle L((x_n)) := \sum_{n=1}^{\infty} \frac{x_n}{\sqrt{n(n+1)}}$.
Find the norm of L.
1
vote
1answer
57 views
Could Ky-Fan Norms Improve the Bound on the Max Norm of $A$
Let $A$ be real symmetric and $D$ shall contain the eigenvalues of $A$. I've learned that $\|A\|_{\text{max}}< \|D\|_{\text{max}}$, where $\|A\|_{\text{max}}$ means the Max norm.
I want to get a ...
1
vote
2answers
42 views
what does it mean to say a space is norm separable?
I came across in my textbook the term: norm separable. I looked in the textbook and online and could not find a definition.
0
votes
0answers
43 views
Expectation inequality with weight matrix
Given that $X$ is random matrix ($X'$ is transposed matrix) and $\Omega$ is invertible square matrix is inequality
$$EX'X\cdot(EX'\Omega X)^{-1}\cdot EX'X\leq EX'\Omega^{-1}X$$
correct? All the ...
1
vote
1answer
28 views
How are $\|A\|_{\text{max}}$ and $\|D\|_{\text{max}}$ related?
Max norm
The max norm is the elementwise norm with $p = \infty$:
$$
\|A\|_{\text{max}} = \max \{|a_{ij}|\}. $$
This norm is not sub-multiplicative.
Let $A$ be real symmetric and $D$ ...
1
vote
1answer
85 views
essential supremum of a matrix multiplication operator
Suppose we have the space $L^p(R,R^n)$ where $1 \leq p < \infty$ (i.e the space of functions that take values in $R^n$ and are $L^p$ integrable) and suppose $T_m: L^p(R,R^n) \to L^p(R,R^n) $ is a ...
2
votes
1answer
91 views
Norm of operator $g\mapsto \int fg$
Let $T_f:(C([a,b],\mathbb C), \lVert \cdot \rVert_1) \to \mathbb C$ with $g\mapsto \int_a^b f(x)g(x) dx$ for any given $f\in C([a,b],\mathbb C)$. I have to find the norm of $T_f$. I started with:
...
1
vote
1answer
174 views
Gradient of squared Frobenius norm
I'd like to find the gradient of $\frac{1}{2} ||X A^T||_F^2$ with respect to $X_{ij}$.
Going by the chain rule in the Matrix Cookbook (eqn 126), it's something like
$\partial \left[\frac{1}{2} ||X ...
1
vote
3answers
100 views
Proving $|x|$ is a norm in $\mathbb {R}^n$
I need some direction on how to start on showing that $| x+y|\leq|x|+|y|$ in $\mathbb R^n$.
Note that
$$
|x|=\left(\sum\limits_{j=0}^n x_i^2\right)^{1/2}
$$
Thank you,
Klara
6
votes
1answer
398 views
Trace Norm properties
Let $||A||_1=\operatorname{trace}(\sqrt{A^* A})$. I already proved that for arbitrary unitary matrices $U$ and $V$, $||UAV^*||_1=||A||_1$ and $||A||_1=\sigma_1+\dots+\sigma_k$. Now I would like to ...
1
vote
1answer
79 views
The openness of the set of positive definite square matrices
Let $\mathbb{R}^{n\times n}$ be the vector space of square matrices with real entries.
For each $A\in \mathbb{R}^{n\times n}$ we consider the norms given by:
$$
\displaystyle\|A\|_1=\max_{1\leq j\leq ...
1
vote
2answers
80 views
Help with an operator norm
Let $T\in \ell_\infty(\mathbb{Z,\mathbb{C}})^*$ such that:
$T(1_{\ell_\infty})=1$ where $1_{\ell_\infty}$ denotes the constant function $1$;
$T(u)\geq 0$ whenever $u$ is real positive.
How to ...
