All Questions

Given a normed space $X$, a closed proper subspace $Y$ and $\alpha\in (0,1)$, the Riesz Lemma states that there is $x\in X$ such that $\|x\|=1$ and $d(x,Y)>\alpha$. Observe that also $d(-x,Y)=d(x,Y)...

The following is essentially a partial case for my previous question. Let $B\subset\mathbb{R}^m$ be the unit ball with respect to a concrete norm on $\mathbb{R}^m$, say $l^p$-norm, $p\in (1,\infty)$....

For $1 \le p \le q \le \infty$, I need an inequality bounding the $\ell_q$ norm from above by the $\ell_p$ norm on $\mathbb{R}^n$: finding a $\lambda$ so that $$ \Vert v \Vert_q \le \lambda \Vert v \...

In ${\bf M}_n(\mathbb R)$, let us consider the usual operator norm $$\|A\|=\sup\frac{\|Ax\|}{\|x\|},$$ where $\|x\|$ is the Euclidian norm. The closed unit ball $B$ is the set of contractions (in the ...