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8
votes
0answers
434 views

Bounding sum of first singular values squared for Kronecker sum of traceless matrices

Let $A$ and $B$ be $4\times4$ traceless matrices with Hilbert-Schmidt norms summing up to $1/4$, i.e. $$\text{Tr}\left[ A\right]=\text{Tr}\left[ B\right] = 0,\qquad\text{Tr}\left[ A^\dagger A + ...
14
votes
4answers
511 views

Bounding the commutator [A,B] in terms of the numerical radius

Given a norm $N$ over ${\bf M}_n(\mathbb C)$, it is a natural question to find the best constant $C_N$ such that $$N([A,B])\le C_N N(A)N(B),\qquad\forall A,B\in{\bf M}_n(\mathbb C).$$ The answer is ...
0
votes
0answers
338 views

Surjectively isometric normed spaces: Hamel vs (extended) Schauder dimension

Bonjour/bonsoir à toutes et à tous. This may really be a very basic question, but... Let $\mathbf{X} \equiv (X, \|\cdot\|_X)$ and $\mathbf{Y} \equiv (Y, \|\cdot\|_Y)$ be surjectively isometric (1) ...
3
votes
2answers
403 views

“Measuring” how far is one Banach space from being surjectively isometric to another

Bonjour/bonsoir à toutes et à tous. Assume that $\mathbf{V} \equiv (V, \|\cdot\|_V)$ and $\mathbf{W} \equiv (W, \|\cdot\|_W)$ are Banach spaces (over the real or complex field). Question 1. What ...
0
votes
1answer
152 views

Truncated Dirichlet series take their supremum on the imaginary axis

Hi there, I am struggling with a theorem about truncated Dirichlet series. I am trying to prove the following theorem: Let $(a_n)_n \subset \mathbb{C}$ and $N \in \mathbb{N}$. Then $\sup_{t \in ...
3
votes
3answers
746 views

Minimum norm of convex hull

Dear all, I am currently stuck at a problem which seems too easy to be stuck at to me... Summary Let $H$ be the convex hull of the points $d_1,\ldots, d_n\in \mathbb{R}^d$. How can one compute ...
2
votes
1answer
555 views

When do 0-preserving isometries have to be linear?

Let $\langle \mathbf{V},+,\cdot,||.|| \rangle$ be a normed vector space over $\mathbb{R}$. Let $f : \mathbf{V} \to \mathbf{V}$ be an isometry that satisfies $f(\mathbf{0}) = \mathbf{0}$ . What ...
3
votes
0answers
325 views

Bounding linear combination of exponentially decaying functions.

Let $f(x)=e^{-a|x|}$ and $a>0$ a constant. Suppose we take a linear combination $s(x) = \sum_{i=1}^n \alpha_i f(x-x_i)$ where $\alpha_i\in\mathbb{R}$ and $-r< x_1< \ldots < x_n< r$. ...
3
votes
1answer
672 views

Inequalities and bounds for relating p-norms (Reference request)

Hello all, I'm trying to find a good resource for a discussion on the relation between say, the p-norm of a vector (from a finite dimensional vector space) and its Euclidean norm. In my search on the ...
2
votes
2answers
266 views

Analogue of an orthogonal subspace in a noneuclidian normed space

This question is related to http://mathoverflow.net/questions/50600/an-existence-question-on-linear-map. If the answer to this question is yes, it would solve the abovementioned other MO question. We ...