Tagged Questions
Norm is a function on a vector space $X$ which generalizes notion of length of vector in general vector spaces.
5
votes
0answers
210 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
0answers
47 views
Equivalence between the matrix 2-norm and a generalized Frobenius “norm” for block partitioned matrices
Recently I had the problem to find an estimate of a $2$-norm of a block partitioned matrix
$$
A = \begin{bmatrix}
A_{11} & \cdots & A_{1n} \\ \vdots & \ddots & \vdots \\ A_{n1} ...
4
votes
0answers
73 views
What are norms used for?
These two questions are quite similar to this one, so I apologise if this irritates anyone. Also, I suspect that a lot can be said in the answer, so I am really just looking for some main points ...
3
votes
0answers
42 views
+100
When does $\|z^2\|=\|z\|^2$
Let $k \in \mathbb{Z}$ and consider the field extension $K := \mathbb{Q}[\sqrt{k}]$. Define a norm on $K$ given by $\|p+q\sqrt{k}\| := \sqrt{p^2+q^2}$. For any $z \in K$, I was interested to know when ...
3
votes
0answers
186 views
Prove this equality in a Hilbert Space $H$ with Riesz's Representation Theorem.
For $x \in H$, prove that $\sup_{\|z\| = 1} \langle x , z \rangle= \| x \|$
Here is a quick one. If someone could improve this it would be great
Proof
By Cauchy Schwarz, $\langle x,z \rangle ...
3
votes
0answers
95 views
The norm of an operator
Let $\rho(x)$ be a weight function in a unit sphere, such that
\begin{equation}
\begin{array}{l}
\displaystyle 1. \rho(x)\ge 0,\int_{\mathbb{R}^n}\rho(x)=1\\
\displaystyle 2. \rho(x)\in ...
3
votes
0answers
59 views
Exercise from textbook about norm
The below diagram included a relevant page and the question 4 of the exercise which i am not sure how to do. Any hint or solution are welcome
Attempts: a) I have done it
b) i have tried to show that ...
3
votes
0answers
97 views
Absolute norms and 1-unconditional sums
Absolute norm
Let $X$ and $Y$ be Banach spaces. Let $Z=X\times Y$ a norm $\|\cdot\|_N$ on $Z$ is called absolute if there is a function $N\colon R^2\rightarrow R$ such that
$$
\|(x,y)\|_N=N((\|x\|, ...
3
votes
0answers
81 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 $|a| \neq 0$, ...
3
votes
0answers
79 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 ...
3
votes
0answers
153 views
Minimizing maximum absolute column sum norm of the residual between a matrix and its $k$-rank approximation
Let $X \in \mathbb{R}^{m\times n}$ be a matrix with rank $r$.
How can we find the optimal $\tilde{X} \in \mathbb{R}^{m\times n}$ whose rank is $k$ where $k\leq r$ and the reconstruction error in ...
2
votes
0answers
33 views
Proving norm equivalence in $W^{1-1/p,p}(\Omega)$
Define for $p\in [1,\infty)$ and $\Omega=(0,1)^N\subset\mathbb{R}^N$ $$W^{1-1/p,p}(\Omega)=\left\{u\in L^p(\Omega): \ \int_\Omega\int_\Omega\frac{|u(x)-u(y)|^p}{|x-y|^{N-1+p}}dxdy<\infty\right\}$$
...
2
votes
0answers
88 views
Operators from $L^{\infty}$ to $L^{\infty}$, below bound of the norm
If $T(f)(x)=\int K(x,y)f(y)dy$, where $K(x,y)$ is locally integrable, is bounded on $L^{\infty}$, how can we show that $\|\int|K(\cdot,y)|dy\|_{L^{\infty}}\le \|T\|_{L^{\infty}\rightarrow ...
2
votes
0answers
40 views
Find a matrix that minimizes the norm of a linear function
Let $S=AA^{T}+XX^{T}-AB^{T}\left(BB^{T}\right)^{-1}BA^{T}$ where $A$ and $B$ are known.
What is the best possible $X$
that minimizes $\left\Vert S^{-1}y\right\Vert _{2}$
for any vector $y\in ...
2
votes
0answers
37 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. ...
