Norm is a function on a vector space $X$ which generalizes notion of length of vector in general vector spaces.
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1answer
75 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
108 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 ...
0
votes
1answer
53 views
Inequality for singular value for differences of matrices (upper bound)
Does anybody know the inequality of singular value for differences of matrices, i.e.
$\sigma_{max}\left(\begin{array}{c}
A-B\end{array}\right)\leq??$
in term of $\sigma_{max}\left(\begin{array}{c}
...
5
votes
2answers
119 views
Calculate $\left\Vert \begin{bmatrix}1 & 2\\ 2 & 4 \end{bmatrix} \right\Vert$
With $$\left\Vert A \right\Vert=\max_{\mathbf{x}\ne 0}\frac{\left\Vert A\mathbf{x}\right\Vert }{\left\Vert \mathbf{x}\right\Vert }$$ and $$A=\begin{bmatrix}1 & 2\\
2 & 4
\end{bmatrix}
$$
...
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
$$
\|(x,y)\|_N=N((\|x\|, ...
0
votes
1answer
128 views
Is there any main method for finding norm of function in $L_1$ space?
Is there any main method for finding norm of function in $L_1$ space? For example :
$f(x)$ = $\sin x$ in space $L_1[-\pi,\pi]$
1
vote
1answer
53 views
Single norm criterion
Let $E$ be a metrizable locally convex space whose topology is defined by an increasing sequence $\{p_n\}$ of seminorms. Show that the topology of $E$ can be defined by a single norm iff there ...
2
votes
1answer
143 views
Upper and lower bounds of a ratio involving vector norms
I'm working on a signal processing problem and need to analyze the following expression
$$
G = \frac{n}{\sum\limits_{i=1}^n |w_i|} \frac{ \sum\limits_{i=1}^n g_i w_i^2}{\sum\limits_{i=1}^n g_i |w_i|}
...
1
vote
2answers
70 views
Matrix norm less than $1$ iteration
Is the following true always for a matrix norm
$$\lVert AB\rVert \leqslant \lVert A\rVert \cdot \lVert B\rVert \text{ ?}$$
Related to this
given $r$ is positive constant, $H$ is symmetric positive ...
2
votes
1answer
35 views
Prove that absorbing sets induce quasinorms
I have been trying to prove this for so long that I pose this to Math.SE:
Let D be a bounded (in regards to, for example, the maximum norm), absorbing subset of a finite-dimensional vector space V ...
1
vote
0answers
82 views
Why is the subdifferential of norm of a matrix ||A|| defined like this?
I read in a paper called "Characterization of the subdifferential of some matrix norms"
that it defines the subdifferential of the matrix norm like this:
$$\partial ||A||=\{G \in R^{m\times n} : ...
1
vote
1answer
68 views
norm of a matrix ( which norm have to use ?)
I need to find the norm of the matrix
$$
A=\left( \begin{array}{cc} e^{-x} \cos( \sin x) & e^{-x} \sin ( \sin x) \\
-e^{-x} \sin ( \sin x) & e^{-x} \cos (\sin x) \end{array} \right)
$$
Here ...
3
votes
0answers
75 views
Is $\rho(f) := \mu(|f| \geq \epsilon), \epsilon >0$ some kind of generalized norm?
Let $\rho(f) := \mu(|f| \geq \epsilon)$ where $\epsilon >0$ is a constant and $\mu$ is a measure and $f$ is a measurable function. Then
$f\equiv 0 \rightarrow \rho(f) = 0$
when $|a| \neq 0$, ...
1
vote
1answer
34 views
Question about norms
Denote supnorm by $|x| = \max\{|x_1|,...,|x_n|\}$ where $x \in \mathbb{R}^n$. How can we show that this norm and Euclidean norm satisfies the following inequality?
$$ |x| \leq ||x|| \leq ...
1
vote
1answer
64 views
Small question regarding norms and Holder conjugates.
I'm trying show that if $p,q$
are Holder Conjugates then: $$\forall\, a\in\mathbb{R}^{n}:\,\Vert a\Vert_{q}=\max_{x\in\mathbb{R}^{n},\,\Vert x\Vert_{p}=1}\left<a,x\right>$$
Where ...
3
votes
1answer
190 views
Matrix norm inequality implying eigenvector norm inequality
For a matrix $A$ let $\|A\|$ be the norm given by $\|A\|=\sup_{v \neq 0}\frac{\|Av\|}{\|v\|}$ where $\|v\|$ is the Euclidian norm on the vector $v$.
Suppose we have matrices $M$ and $S$ with leading ...
3
votes
4answers
148 views
The difference between $L_1$ and $L_2$ norm?
I have been trying to understand what is the difference between $L_1$ and $L_2$ norm and cant figure it out.
In this webpage I got a clear understanding of why we would use $L_1$ norm (scroll down ...
4
votes
5answers
67 views
Norm of vector in $\mathbb{R}^3$ with multiple
If we have a vector in $\mathbb{R}^3$ (or any Euclidian space I suppose), say $v = (-3,-6,-9)$, then:
May I always "factor" out a constant from a vector, as in this example like $(-3,-6,-9) = ...
1
vote
1answer
64 views
Matrix norm relationship
Suppose we have two matrices, A and B, and
$\left\Vert A\right\Vert _{F}\geq\left\Vert B\right\Vert _{F}$
where $\left\Vert .\right\Vert _{F}$ denotes Frobenius norm. Does it imply
$\left\Vert ...
3
votes
2answers
99 views
Linear Algebra munkres analysis on manifolds question.
If $A$ is an $n$ by $m$ matrix and $B$ is an $m$ by $p$ matrix, then
$$ |AB| \leq m|A||B|$$
where $|A| = \max\{|a_{ij}| : i = 1,\ldots,n \text{ and} j = 1,\ldots,m\}$
Attempt: $ |AB| = \max\{| ...
5
votes
1answer
108 views
Norm in a dual space
If $f \in X^*$, with $X^*$ the dual space consisting of all linear bounded functionals on a linear normed space $X$. With the norm defined as $||f||_{X^{*}} = \sup_{||x|| \leqslant 1} |f(x)|$. Why ...
7
votes
3answers
169 views
Attaining the norm of an ideal in a number field by the norm of an element
Let $K$ be a number field of degree $n$ and $\mathfrak{a}$ be an ideal in its ring of integers $\mathcal{O}_K$. We can consider:
The norm $N(\mathfrak{a})$ of $\mathfrak{a}$.
The norms $N(x)$ of the ...
2
votes
2answers
48 views
Why the max can be found only with normalized vectors?
Can someone explain why the term in the first {} equals the second term in the {} :
1
vote
2answers
182 views
Proof of Clarkson's Inequality
Trying to find a proof for Clarkson's inequality, which states that if $2 \leq p < \infty$, then for any $f, g \in L^p$, we have that $$\left|\left|f+g\right|\right|_p^p + ...
1
vote
0answers
57 views
How to represent and handle eigenvalue constraint in optimization
For example consider the problem
$\min_X f(X)$
s.t. $\lambda_i(X+A)=\lambda_i(B)$ for $i \in {1,...,N}$
where $A$ and $B$ are full rank N by N matrix, $\lambda_i(X)$ is the i-th eigenvalue of $X$
...
2
votes
2answers
69 views
is product of norms convex?
Is a function of the form
$f(x) = \|x\|_1\|x\|_2$
convex in x? I have tried plotting it in wolfram alpha and it appears convex, althought I ahve not been able to show it yet
2
votes
2answers
410 views
convergence of $L^p $ norm [duplicate]
Possible Duplicate:
Limit of $L^p$ norm
If I define $|f|_{L^\infty}= \lim_{n\to \infty} |f|_{L^n}$. How can I prove that this limit is esssup $|f|$?
1
vote
2answers
54 views
Norm properties and completeness
Let $(X,||.||_X)$ be a normed space, M,N two subspaces with norms $||.||_M,||.||_N$
The identity maps are cont. Now I can define the norm $||x||_{M+N}=inf\{||m||_M+||n||_N:m\in M, n\in N, x=m+n\}$
...
2
votes
1answer
86 views
Quotient norm on $X\backslash M$
I have $X=(C([0,1]),||.||_1)$ where $||f||_1=\int_{0}^{1}|f(t)|dt$ and $M=\{f\in C([a,b]): f(0)=0\}$. Now I have three questions:
1) Is the quotient norm a norm on the quotient space X\M ?
What I ...
1
vote
1answer
86 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
53 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
77 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
197 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)$$
It makes sense that the 2-norm would be less than or equal to the frobenius norm but I dont know how ...
2
votes
0answers
118 views
Equivalence of norms is a equivalence relation
Two norms $||-||_1 $, $||-||_2$are equivalent if:
for two constants $a,b$ and $x$ from $V$ a vector space over a field it holds that : $$a||x||_1\leqslant ||x||_2\leqslant b||x||_1.$$
This is a ...
2
votes
1answer
163 views
Does the limit of a convergent sequence depend on the norm?
Let $X$ be a vector space, and $\|\cdot\|_1$ and $\|\cdot\|_2$ two different (non-equivalent) Norms on $X.$ Let $(x_n)\subset X$ be a sequence and $x\in X$ such that $\lim_{n\to\infty}\|x_n-x\|_1=0.$ ...
2
votes
3answers
100 views
Proving a matrix equality
I have 2 matrices: $A \in R^{nxn}$ is a non-singular matrix and $B \in R^{nxn}$ is a singular matrix. Here is the expression I need to prove:
$$||A - B|| \ge ||A^{-1}||^{-1}$$
I dont understand why ...
1
vote
2answers
284 views
prove matrix norm equivalence
Given $A \in R^{m\times n}$, I need to prove:
$$||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
107 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
227 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
112 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
41 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 ...
3
votes
2answers
175 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
51 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 $$A=
\begin{bmatrix}
1 & 0.999 \\
...
5
votes
1answer
79 views
relation between norms of two vectors
Can we say that;
if the $l_1$-norm of an arbitrary vector $a$ is smaller that $l_1$-norm of $b$ ($||a||_1 \le ||b||_1$) then the $l_2$-norm of $a$ is smaller than $l_2$-norm of $b$
($||a||_2 \le ...
1
vote
1answer
39 views
Determine operator norm of mutiplication operator
Consider
$$T: (C[-1,1],\|\cdot\|_{2})\rightarrow \mathbb{C}\\Tf :=\int_{-1}^{1}mf\,\mathrm{d}x$$ where $m\in C[-1,1]$.
I want to prove $\|T\| = \|m\|_2$.
$\|T\|\leq\|m\|_2$ can be easily proved by ...
0
votes
0answers
56 views
on norm of submatrix of the inverse and inverse of a submatrix
Given an M-matrix, say $M\in\mathbb{R}^{n\times n}$, which in block form is
$M$ = \begin{pmatrix}
A & B\\
C & D
\end{pmatrix}
where $A\in\mathbb{R}^{k\times k}$ and ...
2
votes
1answer
86 views
Equivalence of two norms
Define two norms as following: $$
\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
304 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
70 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) = ...
