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26 votes
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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 $\mathbb{R}^n$ ...
Jochen Glueck's user avatar
25 votes
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Hölder's inequality for matrices

There are (at least two) "generalizations" of Hölder inequality to the non-commutative case. One is the so called tracial matrix Hölder inequality: $$ |\langle A, B \rangle_{HS} |= |\mathrm{...
lcv's user avatar
  • 506
16 votes

Operator norm of square root of matrix vs original

This is not true. For example, if $$ A = \frac{2}{1 + \sqrt{5}}\begin{bmatrix} 1 & 1 \\ 0 & 1\end{bmatrix} $$ then $\|A\| = 1$. Furthermore, $A$ has exactly $2$ square roots, which are $$ ...
Nathaniel Johnston's user avatar
16 votes

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 ...
Philip Brooker's user avatar
14 votes
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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 ...
Fan Zheng's user avatar
  • 5,139
13 votes

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

There's a very simple approach in finite dimension. Let $G$ be the linear isometry group of $(\mathbf{R}^n,\|\cdot\|_p)$, $1\le p\le\infty$. Let $W$ be the group of signed permutations. First, since $...
YCor's user avatar
  • 61.6k
11 votes
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Subtracting the weak limit reduces the norm in the limit

The property you indicate is known as (strict) Opial’s Property (see https://en.m.wikipedia.org/wiki/Opial_property). It fails generally in reflexive spaces; in fact, it fails generally even for ...
Jack L.'s user avatar
  • 1,443
10 votes

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

By Montel's theorem, every bounded set (w.r.t. the topology of uniform convergence on compact sets) in the space of holomorphic functions is relatively compact. If the space were normed its closed ...
Jochen Wengenroth's user avatar
10 votes
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Seminorm which is zero on dense subset

It depends on whether $\hat{X}$ spans $X$ (in the algebraic sense, i.e. finite linear combinations). If it does, then for every $x \in X$, we can write $x = a_1 x_1 + \dots + a_n x_n$ for some $x_1, \...
Nate Eldredge's user avatar
10 votes

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 ...
Mikhail Ostrovskii's user avatar
10 votes
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 ...
Bill Johnson's user avatar
  • 31.1k
10 votes
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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 $M_n$ ...
Nik Weaver's user avatar
  • 42.4k
9 votes

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 (...
Loïc Teyssier's user avatar
9 votes
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 ...
Suvrit's user avatar
  • 28.4k
9 votes

Bounding supremum norm of Lipschitz function by L1 norm

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 ...
Iosif Pinelis's user avatar
9 votes

Embedding of a Banach space into a Hilbert space

As mentioned in my comment, this is true for reflexive Banach spaces and the compactness game may generalize to other situations, e.g., if the Banach space is a dual space and the embedding in $\sigma^...
Jochen Wengenroth's user avatar
9 votes
Accepted

Banach space with uncountable basis

I will pull together the comments into a community wiki answer with some of my own remarks so that the question isn't left on the unanswered questions list. If you're willing to accept that it is ...
8 votes
Accepted

Non-equivalent norms on finite dimensional vector spaces over a non-complete field

Field $\mathbb Q$ with the usual absolute value $|\cdot|$ from the real numbers. Two norms on $\mathbb Q^2$ ... $$ \|(x,y)\|_1 = |x|+|y| $$ and $$ \|(x,y)\|_2 = \left|\,x+\sqrt{2}\;y\,\right| $$ In ...
Gerald Edgar's user avatar
  • 40.6k
8 votes
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Image of the norm map for degree $3$ galois extension over $\mathbb{Q}$

There is a general strategy to tackle such question. Since the norm is multiplicative, it make sense to look for a prime $p$ not in the image. Then, we can ask about the number of times $p$ divides a ...
S. carmeli's user avatar
  • 4,074
8 votes
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Norms as Points in $C(X)$

$\newcommand\Abs[1]{\lVert{#1}\rVert}\newcommand\abs[1]{\lvert#1\rvert}$If one also enforces non-triviality and continuity, then there is a one-to-one correspondence between multiplicative seminorms ...
Terry Tao's user avatar
  • 111k
8 votes

Norms as Points in $C(X)$

A slight improvement to Terry Tao's answer: It is not necessary to assume a priori that the multiplicative seminorm $\|\cdot\|$ is continuous with respect to the sup-norm on $C(X)=C(X\to \mathbb R)$ ...
Jochen Wengenroth's user avatar
8 votes
Accepted

Estimate of Hölder Norms (Littlewood–Paley theory)

Let $a:=\alpha$, $[h]_a:=[h]_{C^a}$, and $\|h\|_\infty:=\|h\|_{L^\infty}$. For any distinct $x$ and $y$, \begin{align*} |f(x)g(x)-f(y)g(y)|&=|f(x)g(x)-f(y)g(x)+f(y)g(x)-f(y)g(y)| \\ &\...
Iosif Pinelis's user avatar
8 votes
Accepted

Lp norm of Hadamard matrix

Important Edit: As J.J Green pointed out, the OP contains an incorrectly stated value for $\|H\|_{\infty}$, which I copied without checking below. Interpolating between $(1,\infty)$ using the ...
Willie Wong's user avatar
  • 38.4k
8 votes

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

Nik Weaver's answer gives a nice counter-example. Let me just say a few words of context. Kernels $K$ for which $KT$ is trace-class for all trace-class $T$ are called Schur multipliers. (Not to be ...
Matthew Daws's user avatar
  • 18.5k
8 votes
Accepted

prove with a probability of at least $1/e$: $\left\|\sum_{i=1}^k a_{i} P_{i}\right\|_2 \geq\left\|P_{1}\right\|_2$ holds

$\newcommand\si\sigma$Let $$p:=P\Big(\Big\|\sum_{i=1}^k a_i P_i\Big\| \ge\|P_1\|\Big).$$ Here $\|\cdot\|:=\|\cdot\|_2$. The inequality $p\ge1/e=0.367\dots$ does not hold in general. Actually, the best ...
Iosif Pinelis's user avatar
8 votes

What norms can be "universally" defined on any real vector space with a fixed basis?

What you call the "usual $\ell^p$ norm on $V$" is anything but when $V$ is infinite-dimensional. For example when $p=2$, your norm is (as in the standard setting) induced by the scalar ...
Christian Remling's user avatar
7 votes

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

Concerning your first question, as I noted in my comment above for any operator norm we have $\Vert [A,B]\Vert \leq 2\Vert A\Vert\Vert B\Vert$. Conversely, let $A = \pmatrix{1 & 0 \\ 0 & -1}$,...
Lior Silberman's user avatar
7 votes
Accepted

Regarding spectral radius

Yes, see T. Ando, On hyponormal operators, Proc. Amer. Math. Soc. 14 (1963), 290-291. His main result states that $\|T^n\| = \|T\|^n$ for any hyponormal operator, which implies the conclusion by ...
Nik Weaver's user avatar
  • 42.4k
7 votes
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How to calculate or estimate RKHS norm?

For a concise introduction to RKHS, you could have a look at sections 2.3 and 2.4 of Gaussian Processes and Kernel Methods: A Review on Connections and Equivalences by Kanagawa et al. (2018). In ...
Cédric Travelletti's user avatar
7 votes
Accepted

Bounding supremum norm of Lipschitz function by L1 norm

$\newcommand\Om\Omega$Now consider the general case of any natural $d$. Here we will give an upper bound on $\|f\|_\infty$ in terms of $\|f\|_1$, $L$, and $d$. This bound will be optimal up to a ...
Iosif Pinelis's user avatar

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