bio  website  math.tamu.edu/~johnson 

location  Texas A&M  
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visits  member for  5 years, 4 months 
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Distinguished Professor of Mathematics at Texas A&M University
2d

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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 
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SubGROUPs of Banach spaces, when are they dense in a vector subspace?
Yes, their Acta paper. 
Apr 27 
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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 
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A question involving Mazur's Lemma
The BanachSaks 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 
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Are “most” operators on an infinitedimensional complex Banach space “diagonalizable”?
The ArgyrosHaydon spaces I mentioned have Schauder bases, so the finite rank operators are dense in the compact operatorsI 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 
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Are “most” operators on an infinitedimensional 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 
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On the modulus of convexity of mixednorm $\ell_{p_1,p_2}$ 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 
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On the modulus of convexity of mixednorm $\ell_{p_1,p_2}$ 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 LindenstraussTzafriri. I think it contains what you need; at least it will refer to the relevant papers. 
Apr 2 
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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 
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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 
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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. 
Apr 1 
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How large can a set of nearly equidistant points be?
$D$ can be exponential in $k$. This follows, for example, from the JohnsonLindenstrauss lemma, which you can find via Google. 
Mar 30 
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Max min of functionals
I do not understand your claimed lower bound. For fixed $m$ it should be increasing in $n$; moreover, it is clearly wrong for $n=1$. 
Mar 30 
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Approximation property of Fréchet if range is restricted to an embedded Hilbert space
No. For any separable, infinite dimensional Banach space $X$, there is a one to one compact operator from $\ell_2$ into $X$ that has dense range. This is a simple exercise. 
Mar 30 
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Approximation property of Fréchet if range is restricted to an embedded Hilbert space
No, not even if $W$ is a Banach space. What you want implies that $W$ has the bounded approximation property. 
Mar 30 
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Is the conditional expectation a contraction in weak $\mathbb L^p$ spaces?
The CalderonMitiagin interpolation theorem says that all rearrangement invariant function spaces are exact interpolation spaces between $L_1$ and $L_\infty$. This is Theorem 2.2 in BennettSharpley and can be found in many other books as well. 
Mar 25 
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reflexive banach space
Thanks, Tim; I corrected. 