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This question is more or less a cross post of https://math.stackexchange.com/questions/1218660/orthogonal-functions-with-shrinking-support. Let $X$ be a metric space (compact, if it helps) and let $Y …

As discussed in the comments, the statement probably needs to be modified in order for $\langle D \rangle^{-n}$ to be defined. I'm guessing that the correct statement should fit into the following fr …

I think Helge's answer cuts to the historical heart of the matter: solution operators for various differential equations tend to be bounded, non-compact operators (obtained in many cases from an unbou …

I originally posted this question on math.stackexchange (https://math.stackexchange.com/questions/83182/modulus-of-continuity-take-2), but it's been a few days and I haven't received any correct answe …

I have posed this question to some experts at my university who would probably know the answer if there were a complete one, so my expectations are limited. It's possible that the question deserves t …

Let $H$ be a separable Hilbert space and let $\{e_i\}$ be an orthonormal basis. My first question is: Is there a probability measure on $B(H)$ such that for $T$ chosen uniformly randomly the matr …

I have to confess to being more confused by the theory of pseudodifferential operators than I should be, but I think an answer to a question at least related to yours is briefly discussed in chapter 2 …

Given any first order, graded, self-adjoint, elliptic operator on a manifold the spectrum is unbounded in both the positive and negative directions. There are probably elementary ways to see this, bu …

I imagine most people who frequent MO have been indoctrinated into the point of view that the Riemann integral can be safely discarded once one has taken the time to develop the Lebesgue integral. Af …

Maybe it will help to see how the algebra $B$ is a special case of the general construction in $2.\alpha$. The compact manifold is the space $Y = \{a, b\}$ consisting of two points. The open cover o …

I've never completely understood what counts as "an application of category theory". With other areas of mathematics an "application" of area A to area B is generally a result which translates a prob …

EDIT (much later): My answer below is not quite correct, as pointed out by Dan Ramras, because I ignored the importance of base points in the definition of $\pi_n$. See Ramras' answer for the needed …

I don't know the history at all, but I have to imagine that the language was invented to provide a context for talking about solution operators for differential equations. Consider, for example, the …

One of the major geometric applications of the sort of analysis that you describe is to index theory for elliptic operators on manifolds. Using geometry one can often construct a differential operato …

If you're willing to compromise on the operator algebras part then "Introduction to Hilbert Spaces with Applications" is close to optimal. It includes not only a detailed discussion of the spectral t …