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I am a professor of Mathematics at the University of Pisa, Italy.
1d

comment 
Supremum of positve kernel
..and you can apply the finite dimensional result to $P_\pi T P_\pi $, where $P_\pi $ is the orthogonal projector on $L^2$ wrto the subalgebra generated by a finite measurable partition $\pi$. Taking a limit on the partitions you get the stated equality. 
Jan 28 
comment 
Riemann Siegel function and gamma function
Yes, but if a question originates a non trivial and interesting answer, it shouldn't be that bad. Anyway, I respect your sportsmanlike attitude. 
Jan 28 
comment 
Riemann Siegel function and gamma function
I'm curious about this unusual phenomenon (yet expected, by the "Reversal" badge) of +15 points difference between question and answer! Some hints? 
Jan 28 
reviewed  Close Sequence of smooth maps converging to the identity 
Jan 28 
reviewed  Leave Open An integral with respect to the Haar measure on a unitary group 
Jan 28 
comment 
Gateaux but not Frechet differentiable functional
You can see the difference between Gateaux vs Fréchet already in $\mathbb{R}^2$. Since you asked for an example in $L^p$, it is worth recalling that there are very natural examples of nonlinear operators on $L^p$ spaces which are G but not F differentiable. If you need this for your class' needs, you may like to check "A primer on Nonlinear Analysis" by A.Ambrosetti and G.Prodi, a very nice and elementary introduction to Nonl.Anal. 
Jan 28 
reviewed  Leave Open Approximating an arbitrary $\sigma$algebra by simpler $\sigma$algebras 
Jan 28 
reviewed  Leave Open Approximation of quadratic variation 
Jan 28 
reviewed  Leave Open Simplify the product $ I_{\nu}(\delta_1 \sqrt{x}) I_{\mu}(\delta_2 \sqrt{yx}) $ of modified Bessel functions of the first kind 
Jan 28 
reviewed  Leave Open Question on the local existence theory for the classical solution for the incompressible fluid dynamics equation 
Jan 28 
comment 
Sequence of smooth maps converging to the identity
You just want the Inverse Function Theorem, the very classical and elementary one (only, not all statements of it take care of the size of $\epsilon$). Local $C^1$ convergence is enough (while of course just $C^0$ is not). Check e.g. the second application listed here: en.wikipedia.org/wiki/Banach_fixedpoint_theorem#Applications 
Jan 28 
reviewed  Leave Open Lagrange Multipliers for linear functionals 
Jan 28 
reviewed  Leave Open distancesbased dispersion measuring approach 
Jan 28 
comment 
Is there a standard notation for offdiagonal transpose?
Without introducing new notations, this is $PA^TP$, with the matrix $P=[[0,1],[1,0]]$ . 
Jan 27 
comment 
Continuouspiecewiselinear versus piecewiselinear
sorry, I didn't realize it, and in the meanwhile I had some piecewise linear arcs my head. 
Jan 27 
answered  Continuouspiecewiselinear versus piecewiselinear 
Jan 26 
revised 
Legendre transform and Lipschitz approximation
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Jan 26 
revised 
Legendre transform and Lipschitz approximation
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Jan 26 
revised 
Legendre transform and Lipschitz approximation
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Jan 26 
revised 
Legendre transform and Lipschitz approximation
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