# Tag Info

Accepted

### What are the matrices preserving the $\ell^1$-norm?

As pointed out by YCor in the comments, the following theorem is true: Theorem 1 Let $p \in [1,\infty] \setminus \{2\}$. If a matrix $A \in \mathbb{R}^{n \times n}$ is an isometry on $\mathbb{R}^n$ ...
• 11.7k
Accepted

• 5,765

### Cartesian product of Banach spaces: all norms such that the inclusion is an isometry are equivalent?

It is actually not true in general that such a norm $\Vert \cdot \Vert_{\mathcal{A}^n}$ must be complete, despite the fact that the contrary is presented as fact in reputable sources in the literature ...
• 2,283
Accepted

### Can the topological algebra of analytic functions be endowed with a norm that defines the natural topology?

For those who don't have the book (or have the wrong version), here is the proof that the topological vector space of holomorphic functions on the unit disk is not normable (i.e. whose topology is not ...
• 5,139

• 29.5k

### In infinite dimensions, is it possible that convergence of distances to a sequence always implies convergence of that sequence?

It seems to me that you can show that no infinite-dimensional separable Banach space $X$ is P-complete as follows. Pick any bounded separated sequence $\{x_n\}_{n=1}^\infty$ in $X$ and pick a dense ...
• 4,868
Accepted

### In infinite dimensions, is it possible that convergence of distances to a sequence always implies convergence of that sequence?

That every Banach space is contained in a $P$-complete Banach space follows immediately from the following Theorem. Let $X$ be a Banach space. Then there exists a Banach space $Y$ containing $X$ in ...
• 31.1k
Accepted

### If I multiply the coefficients of a trace-class operator with bounded complex numbers is it still trace class?

It's a little more complicated than I thought! Frederik Ravn Klausen pointed out an error. Still, I maintain that the product needn't even be bounded. As the answer to this question shows, in $M_n$ ...
• 42.4k

### Can the topological algebra of analytic functions be endowed with a norm that defines the natural topology?

There is an elementary answer. Let $D$ be any domain of $\mathbb{C}$. The usual derivation operator $\partial : \mathcal{O}(D)\to \mathcal{O}(D)$ is continuous for the topology of uniform convergence (...
• 5,299
Accepted

### Bounding the matrix norm of a commutator $[A,B]$ in terms of the norms of $A$ and $B$

A somewhat more general setting, namely, finding the best constant $C_{p,q,r}$ in \begin{equation*} \|AB-BA\|_p \le C_{p,q,r}\|A\|_q\|B\|_r, \end{equation*} for Schatten $p$,$q$,$r$-norms, is studied ...
• 28.4k
It can be shown that for $d=1$ the best upper bound on $\|f\|_\infty$ is given by $$\|f\|_\infty\le\sqrt{2L\|f\|_1}\,1(\|f\|_1\le L/2)+(L/2+\|f\|_1)\,1(\|f\|_1>L/2).$$ If there is a sufficient ...