Norm is a function on a vector space $X$ which generalizes notion of length of vector in general vector spaces.
1
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2answers
63 views
Proving that Euclidean space having the infinity metric is a complete metric space (stuck)
I am trying to prove that the space ${\mathbb{R}}^k$ with the $\infty$-metric is a complete metric space.
I know that I need to show that every Cauchy sequence in the metric space ${\mathbb{R}}^k$ ...
1
vote
1answer
25 views
$H^1$ function with smallest seminorm
Let $\Omega\subset\mathbb R^d$ be a Lipschitz domain and $u\in H^1(\Omega)$. Find a $w\in H^1(\Omega)$ with the same boundary values but minimal seminorm on $H^1(\Omega)$.
I've read that harmonic ...
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 $|f'(1)-h'(1)|=|g(f)-g(h)|\le \text{max} |f-h|+ \text{max} |f'-h'|$, ...
3
votes
1answer
72 views
Square matrix $\|Ax-Ay\|\le \|x-y\|$
Could you give me an example of a square matrix $A\in \mathcal{M}_{2 \times 2}$ or $\mathcal{M}_{3 \times 3}$ for which we have $\|Ax-Ay\|\le \|x-y\|$, $ \ \ x, y \in \{0, e_1, . . . , e_n\}, \ \ e_1, ...
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.
1
vote
1answer
34 views
Proof of non-strictly convexity of $l_1$ and $l_{\infty}$
Given the Banach space $l_p = \left\{f\in R^n : \|f\| \leq \infty\right\}$ for $1\leq p \leq \infty$, we define the following norms
$\|f\|_p = \left(\displaystyle\sum_{j=0}^{\infty}
...
2
votes
2answers
100 views
+50
Understanding weighted inner product and weighted norms
I am reading this book where at page 27 following definitions about weighted inner product and weighted norms are given.
Let $M$ and $N$ be Hermitian positive definite matrices of order $m$ and ...
1
vote
2answers
184 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 $$\left|\left|f+g\right|\right|_p^p + ...
0
votes
0answers
36 views
Calculation of the sub gradient of the first norm of a matrix
Lets say I have a matrix X and its first norm $||X||_1$. How do I calculate the subgradient of this norm with respect to matrix X itself.
0
votes
1answer
44 views
How to generate a N*D random matrix with columns of unit length?
Is it possible to generate a N*D random matrix with columns of unit length?
If not, I also think it is possible of generating a N*D random matrix and, after that, normalizing it in order to have ...
8
votes
2answers
126 views
Basic question about $\sup_{x\neq 0}{} \frac{\|Ax\|}{\|x\|} = \sup_{\|x\| = 1}{\|Ax\|} $, $x \in\mathbb{R}^n$
I am having trouble with understanding the following definition while studying some basic things related with matrix norms:
For every matrix $A\in M_n(\mathbb{R})$
$$\sup_{x\neq 0}{} ...
0
votes
0answers
19 views
Condition number of a function
I would like to find the Condition number of a function (f(x)) with one variable (x) and several parameters. which can be calculated by:
$$ c(x) = ||x||*||f'(x)||/||f(x)||$$
Here if my function is ...
2
votes
2answers
29 views
Approximation of matrix in 2-norm
The question is the following: Given a matrix $A$ with rank $k$, we are looking for a matrix $B$ of rank $j$, where $j<k$ such that $\|A-B\|_2$ is minimal.
My idea was to choose, if $A=P ...
1
vote
0answers
39 views
why is $\ell_0$ a pseudo-norm?
Let $\mathbf{x}$ be a vector in $\mathbb{R}^n$. We define the $\ell_0$ pseudo-norm by:$$\|\mathbf{x}\|_0=\#\left\{i : \mathbf{x}_i\neq0\right\}$$
Why $\|\cdot\|_0$ is not properly a norm?
0
votes
1answer
33 views
How to expand equation inside the L2-norm?
I want expand an L2-norm with some matrix operation inside.
Assume I have a regression $Y=X\beta+\epsilon$.
I want to solve (meaning expand),
$$\displaystyle\|Y-X\beta \|_{2}^2$$
Should I do:
1)
...
1
vote
1answer
38 views
Is any norm on $\mathbb R^n$ invariant with respect to componentwise absolute value?
Given $\mathbf{x}=(x_1,...,x_n) \in \mathbb{R}^n$ , define $ \mathbf{x}'=(|x_1|,...,|x_n|) $ .
Then, is it $||\mathbf{x}'|| = ||\mathbf{x}||$ for every norm on $ \mathbb{R}^n $ ?
NB: The answer ...
-1
votes
0answers
31 views
matrix norm that is invariant through a change of orthogonal basis
I am looking for (real) matrix norms that would be invariant through a change of orthogonal basis. I know only about the Frobenius norm, which satisfy this condition as it can be expressed as a ...
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.
0
votes
0answers
23 views
L1 penalty can serve as a convex surrogate for an L0 penalty. Why?
I have heard machine learning practitioners say that the $L_1$ penalty is a (or can serve as) convex surrogate for an $L_0$ penalty (in the context of optimization and statistical fitting).
What do ...
0
votes
1answer
84 views
norm of sobolev space $H^{1/2}$
Let $\Omega\subset\mathbb R^d$ a Lipschitz domain and $\Gamma:=\partial\Omega$. For $u\in C^{\infty}(\Gamma)$ we define
$$||u||_{H^{1/2}(\Gamma)} = \inf_{\substack{v\in H^1(\Omega) \\ v|_\Gamma =u}} ...
0
votes
0answers
39 views
Is it general to say “norm” to mean 2-norm when it is on an inner product space?
Let $V$ be an inner product space over $\mathbb{F}$.
If one defines $\lVert \bullet \rVert$ as $\sqrt{\langle \bullet, \bullet \rangle}$, then $\lVert \bullet \rVert$ is a norm on $V$.
However, if ...
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 ...
2
votes
0answers
31 views
Proving norm inequality with Schwarz's inequality
I'm stuck on the following problem:
Suppose that $\{V_j : j \in \mathbb{Z}\}$ is a multiresolution analysis with scaling function $\phi$, and that $\phi$ is continuous and compactly supported. ...
4
votes
1answer
905 views
Equivalent norms
Whenever two norms are equivalent in the sense that $||x||_1\le c_1\cdot ||x||_2$ and $||x||_2\le c_2\cdot ||x||_1$, they generate the same topology. Is the reverse also true, i.e. if a topology is ...
2
votes
1answer
43 views
Need help to simplify an equation
I am computing an error estimate where at the end I got the following term
$\|X_{k} - G\|\leq (q^{2^{k+2}} + q^{2^{k+3}}+ q^{2^{k+4}}....)q^{-3}\|Y_{0}\| + q^{2^{k+1}}.q^{-2}\|X_0\|$ ,
where $X_k$ ...
3
votes
3answers
110 views
Norm and square of the ideal $(2,1+\sqrt{-5})$ in the ring of integers
Let $I=(2,1+\sqrt{-5})$ be an ideal of the ring of integers of $\mathbb Q(\sqrt{-5})$. What is its norm $N(I)$? And is $I^2$ principal?
My notes say:
$1$, $\sqrt{-5}$ is a $\mathbb Z$-basis ...
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 ...
0
votes
0answers
33 views
Subgradient of matrix $l1$-norm
Let $X$ be a square matrix, what is the sub-gradient of $f(X) = ||AX-XB ||_{l1}$?
$A$ and $B$ are both constant matrix.
I am very confuse about the chain rule on matrix derivatives.
5
votes
0answers
130 views
Monotone matrix norms
[Ciarlet 2.2-10]
Let $\mathscr{S}_n$ be the set of symmetric matrices and $\mathscr{S}_n^+$ the subset of non-negative definite symmetric matrices. A matrix norm $\|\cdot\|$ to be monotone if
...
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\} ...
0
votes
1answer
74 views
Difference between maximizing matrix's $l1$ norm and Frobenius norm
The original problem is to maximize the Frobenius norm of a matrix product
$max_{X\in \mathbb{D}}||B^TXA ||_F$, $\mathbb{D}$ is the doubly stochastic matrix set
As this is too difficult due to ...
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
56 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...
9
votes
2answers
180 views
On the convexity of element-wise norm 1 of the inverse
Let us define $\|A\|_1$ the element wise norm 1 of a matrix $A \in \mathbb{R}^{n \times m}$ as
$$
\|A\|_1= \sum_{i,j} |A_{i,j}|.
$$
Obviously, this function is convex over $\mathbb{R}^{n \times m}$. ...
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 ...
1
vote
2answers
54 views
What is the role of supremum in operator norm
An operator norm is defined as $\|A\|_S=\sup\{\|Av\|:v\in \Bbb R^n, \|v\|=1\}$. Where $\|\cdot\|$ is some norm on $\Bbb R^n$ and $A\in M_n(\Bbb F)$, space of square matrices of dimension $n$ over ...
0
votes
1answer
21 views
Linear approximation of matrix norm
Given a square matrix $X=[x_1...x_N]$, and can be vectorized by $y=vec(X)=[x_1^T ... x_N^T]^T$
Is there any linear function can approximate $|| X ||$ (any matrix norm is okay) by using $y$?
1
vote
2answers
260 views
Motivation for normed space definition?
So if $X$ is a vector space, and you define a norm, $x \mapsto \| x \|$, on it, then the bounded subset, $V = \{ x \in X: \|x\| < \infty \}$ is automatically a subspace. This follows from the ...
0
votes
1answer
35 views
Matrix norm proving problem
I have a homework, one of a few, that I should prove that the norm $||A||_S = sup\{||Av||:||v||=1\}$ Where ||.|| is some norm on $R^n$ into which vector space belongs also $v$. I should mention that A ...
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 ...
36
votes
3answers
4k views
Norms Induced by Inner Products and the Parallelogram Law
Let $ V $ be a normed vector space (over $\mathbb{R}$, say, for simplicity) with norm $ || \cdot || $.
It's not hard to show that if $|| \cdot || = \sqrt{\langle \cdot, \cdot \rangle}$ for some ...
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 $V \to \mathbb{C}$.
...
2
votes
0answers
52 views
Differences between a norm in $\mathbb{C}^n$ and a norm in $\mathbb{R}^n$
Consider the definitions of matrix norm and subordinate matrix norm from
Matrix Norm set #2
and let $A$ a real matrix and $\|\cdot\|$ a vector norm over $\mathbb{C}^n$. Define
\begin{eqnarray*}
...
4
votes
2answers
33 views
norm of inverse less than 1
I just wanna ask if what I am doing here make sense:
Let $\epsilon$ be arbitrary positive number. Choosing $\epsilon$ and let it approaches 0, I would like to have $||(I-\epsilon A)^{-1}|| < 1$. ...
4
votes
1answer
80 views
Matrix Norm set #2
As a complement of the question
Matrix Norm set
and in order to complete the Problem 1.4-5 from the book: Numerical Linear Algebra and Optimisaton by Ciarlet. I have this additional conditions:
(3) ...
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 ...
3
votes
2answers
60 views
$q$-norm $\leq$ $p$-norm [duplicate]
[Ciarlet 1.4-8] If $0 < p < q$, show that
$$\left(\sum_{i=1}^n|v_i|^q\right)^{1/q}\ \leq\ \left(\sum_{i=1}^n|v_i|^p\right)^{1/p}$$
Somebody knows how prove that? Thanks in adavance for the ...
7
votes
1answer
220 views
Matrix Norm set
I need help with this problem:
Let $\|\cdot\|$ and $\|\cdot\|^{\prime}$ two matrix norms, and consider the relation
$$\|\cdot\| \leq \|\cdot\|^{\prime}\ \Leftrightarrow\ \|A\| \leq \|A\|^{\prime},$$
...
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 ...
2
votes
1answer
53 views
Two inequalities related to norm
We have some difficulties in the following problem:
Let $H$ be a real Hilbert space.
Find $\alpha>0$ such that
$$
\langle\frac{u}{\sqrt{\|u\|}}-\frac{v}{\sqrt{\|v\|}}, u-v\rangle\geq
...
