6
votes
2answers
310 views

$\|T\|_2 \le \sqrt{\|T\|_1\|T\|_\infty}$

Let $T$ be a linear operator acting on a finite-dimensional real or complex vector space. As a direct consequence (or rather a particular case) of the Riesz-Thorin theorem, we have $$ \|T\|_2 \le ...
0
votes
1answer
154 views

Constructing a continuous matrix valued function

Given $d<k$. Let ${\cal M}_{d\times k}(\mathbb{R})$ denotes the set of all $d\times k$ real matrices and suppose that $H:\mathbb{R}^k\rightarrow {\cal M}_{d\times k}(\mathbb{R})$ is a continuous ...
0
votes
0answers
172 views

Closed-form expressions for dual norms of real normed vector spaces

Didn't get any biters over at MSE, so I figure this place might be more appropriate... Say that $V$ is a finite-dimensional real normed vector space, where for some $v \in V$ the norm is notated by ...
2
votes
1answer
254 views

Decomposing bilinear forms in Hilbert spaces

You are given a complex Hilbert space $H$ with two equivalent Hilbert space structures $<,>$ and $<,>'$. Define $<,>''=<,> + <,>'$ to be the sum of our two scalar ...
2
votes
2answers
412 views

How do you compute the dual norm of an induced norm on a subspace of a finite-dimensional Lp-normed vector space?

Say you have a finite-dimensional vector space $V$ with an $L^p$ norm on it. In general, the norm induced on a subspace $V_s$ of doesn't have to be another $L^p$ norm, so the unit sphere in $V_s$ ...
6
votes
1answer
402 views

Do real vectors attain matrix norms?

I apologize if the following question ends up being too elementary for this website; I asked it on math.SE a week ago and it remains unanswered. Let $A$ be an $n \times n$ matrix with real entries ...
3
votes
1answer
318 views

Completely bounded maps on Mn

The aim of this question is to collect nice maps on $M_n(\mathbb{C})$ with the following property: $\phi_n:M_n(\mathbb{C})\rightarrow M_n(\mathbb{C})$ with $||\phi||=1$ and ...
2
votes
2answers
271 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 ...
1
vote
1answer
435 views

Is exp(rA) = (exp(A))^r for real r and A in a Banach space?

Is $e^{(rA)} = (e^{A})^r$ when $r \in \mathbb{R}$ and $A$ is an element of a Banach algebra? Clearly if $n$ is an integer, then $e^{(nA)} = e^{A+A \cdots +A} = e^{A}e^{A}\cdots e^{A} = (e^{A})^n$, ...
8
votes
1answer
953 views

Banach-Mazur distance between $\ell^p$-norms

Let $E^n$ be the real or complex space of dimension $n$. If $N$ and $M$ are two norms over $E^n$, and if $A$ is an endomorphism, then $$\|A\|^M_N:=\sup_{x\ne0}\frac{M(Ax)}{N(x)}$$ is an operator norm ...
6
votes
2answers
659 views

Inner products and Norms

Let $f:[n]\times [n] \rightarrow [0,1]$ be a function from pair of integers to the real interval [0.1]. I would like to find sets of complex vectors $X= \{x_i\}$ and $Y=\{y_j\}$ satisfying $x_i\cdot ...