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We also prove the riesz representation theorem, which characterizes the bounded linear functionals on a hilbert space, and discuss weak convergence in hilbert spaces. In some places i follow the book closely in others additional material and alternative proofs are given. Introduction classically, functional analysis is the study of function spaces and linear operators between them

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Much of the material and inspiration came from larry brown’s lectures on functional analysis at purdue university in the 1990s, and some came from my reed thesis 1987 Lax, john wiley & sons (2002), referred to as \lax below Adjoints in hilbert spaces recall that the dot product on rn is given by x y = xty, while the dot product on cn is

Functional, so that by lemma 2.14 we learn that is continuous

Hence is a topological homeomorphism, and since we have assumed it is a vector space isomorphism, we conclude all together it is a tvs isomorphism C y ⊆ y y c ⊆ y x ∈ y c next, for the first property, we want to show that , which is equivalent to. As such it established a key framework for the development of modern analysis. Comments and course information these are lecture notes for functional analysis (math 920), spring 2008

The text for this course is functional analysis by peter d