18,918 reputation
248100
bio website math.tamu.edu/~johnson
location Texas A&M
age
visits member for 5 years, 5 months
seen 19 hours ago

Distinguished Professor of Mathematics at Texas A&M University


May
15
comment Sard's Theorem For Banach Spaces
Look at MathSciNet reviews of papers by Sean Bates to see which ones are relevant for you.
May
12
comment Rearrangments of Fourier series
Maybe, but it is trivial that if $X$ has a basis, then for any $f$ in $X$ there is another basis s.t. $f$ is the first vector of the new basis.
May
12
answered Rearrangments of Fourier series
May
10
comment A question on p-summing operators
My answer says "no".
May
9
revised On $p$-summable sequences with respect to operator ideals
added 11 characters in body
May
9
answered A question on p-summing operators
May
4
comment Uncountability of the set of subsets of $\mathbb N$
This should be on Math.StackExchange instead of MO, but... For each real $x$, consider the set of rationals less than $x$.
Apr
27
comment SubGROUPs of Banach spaces, when are they dense in a vector subspace?
Yes, their Acta paper.
Apr
27
comment SubGROUPs of Banach spaces, when are they dense in a vector subspace?
They can. In fact, in separable $L_1$ predual (in particular, the universal space $C[0,1]$) can be isometrically embedded as a norm one complemented subspace of Aff $K$ for some $K$. This is an old result of Lazar and Lindenstrauss.
Apr
22
comment A question involving Mazur's Lemma
The Banach-Saks theorem is not true for every Banach space. There exist normalized unconditional bases that converge weakly to zero and yet the norm of the average of any n terms has norm at least 1/2.
Apr
21
awarded  Nice Answer
Apr
13
answered Fixed point theorems
Apr
9
comment Are “most” operators on an infinite-dimensional complex Banach space “diagonalizable”?
The Argyros-Haydon spaces I mentioned have Schauder bases, so the finite rank operators are dense in the compact operators--I neglected to mention that. I don't know whether on every Banach space the closure of the diagonal operators contains the compact operators.
Apr
8
comment Are “most” operators on an infinite-dimensional complex Banach space “diagonalizable”?
There are infinite dimensional Banach spaces on which every bounded linear operator is of the form $\lambda I + K$ with $K$ a compact operator. On such a space the diagonal operators are dense.
Apr
5
comment On the modulus of convexity of mixed-norm $\ell_{p_1,p_2}$ spaces
When $p=\log n$, $\ell_p^n$ is $\ell_\infty^n$ up to a constant $C$ (the base of the logarithm). So you are looking at $\ell_\infty^n(\ell_1^m)$ in a $C$-equivalent norm.
Apr
2
comment On the modulus of convexity of mixed-norm $\ell_{p_1,p_2}$ spaces
A fairly easy way of seeing that you get the expected is to take advantage of the fact that in Banach lattices you can calculate the moduli of convexity and smoothness in terms of properties of the lattice ($p$ convexity; $p$ concavity). Look at volume 2 of Lindenstrauss-Tzafriri. I think it contains what you need; at least it will refer to the relevant papers.
Apr
2
comment Between Tietze's and Dugundji's Extension Theorems
Similarly, Q1 has a positive answer for separable Frechet spaces, since they are all homeomorphic to the countable product of lines.
Apr
2
answered Between Tietze's and Dugundji's Extension Theorems
Apr
1
comment How large can a set of nearly equidistant points be?
JL says any $m$ points in a Hilbert space $1+\epsilon$-embeds into a Hilbert space of dimension at most $c\epsilon^{-2} \log m$ for some constant $c$ independent of $m$ and $\epsilon$. That is sharp up to $\log (1/\epsilon)$.
Apr
1
comment How large can a set of nearly equidistant points be?
Apply JL to an orthonormal set in a high dimensional space; all the pairwise distance are the same, so you get almost the same in the low dimensional space.