functional-analysis

I want to find an example of a bijective operator on a Hilbert space $T : \mathfrak{H} \rightarrow \mathfrak{H} $ such that its inverse $T^{-1}$ is not bounded. I have seen examples of non-surjective ...

A measurable field of Hilbert spaces is the exact analogue of a vector bundle over a topological space in the setting of fiber bundles of infinite-dimensional Hilbert spaces over measurable spaces. The original definition is due to John von Neumann (Definition 1 in Neumann). We present here a slightly modernized version, which can be found in many modern sources, e.g., Takesaki. Suppose is a meas…

analysis (differential/integral calculus, functional analysis, topology) metric space, normed vector space open ball, open subset, neighbourhood convergence, limit of a sequence compactness, sequential compactness … … constructive mathematics, realizability, computability propositions as types, proofs as programs, computational trinitarianism A Cauchy real number is a real number that is given as…

analysis (differential/integral calculus, functional analysis, topology) metric space, normed vector space open ball, open subset, neighbourhood convergence, limit of a sequence compactness, sequential compactness … … constructive mathematics, realizability, computability propositions as types, proofs as programs, computational trinitarianism basic constructions: strong axioms further In real ana…

UCLA Mathematics Professor Sorin Popa has been elected to the National Academy of Sciences in recognition for his continued research in functional analysis, especially in operator algebras.  Sorin Popa has led an illustrious career, receiving many awards for his original research. Popa was previously elected to the American Academy of Arts and Sciences in 2013. […] The post Professor Sorin Popa E…

Test functions are how you can make sense of functions that aren’t really functions. The canonical example is the Dirac delta “function” that is infinite at the origin, zero everywhere else, and integrates to 1. That description is contradictory: a function that is 0 almost everywhere integrates to 0, even if you work in extended […] The post Test functions first appeared on John D. Cook .

The celebrated decomposition theorem of Fefferman and Stein shows that every function of bounded mean oscillation can be decomposed in the form   modulo constants, for some , where are the Riesz transforms. A technical note here a function in BMO is defined only up to constants (as well as up to the usual almost […]