Linear Analysis: An Introductory Course
Now revised and updated, this brisk introduction to functional analysis is intended for advanced undergraduate students, typically final year, who have had some background in real analysis. The author's aim is not just to cover the standard material in a standard way, but to present results of application in contemporary mathematics and to show the relevance of functional analysis to other areas. Unusual topics covered include the geometry of finite-dimensional spaces, invariant subspaces, fixed-point theorems, and the Bishop-Phelps theorem. An outstanding feature is the large number of exercises, some straightforward, some challenging, none uninteresting.
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Normed spaces and bounded linear operators
Linear functionals and the HahnBanach theorem
Finitedimensional normed spaces
The Baire category theorem and the closedgraph theorem
Continuous functions on compact spaces and the StoneWeierstrass theorem
The contractionmapping theorem
Weak topologies and Hilbert spaces
Euclidean spaces and Hilbert spaces
analysis assertion assume Banach space basis bounded linear called chapter Check claim Clearly closed subspace closure compact complete concerning consequence consider containing continuous function convergent convex Corollary Deduce define dense differentiable dual easily eigenvalue element equality equicontinuous equivalent Euclidean space example Exercise existence extension fact finite finite-dimensional fixed point function f Furthermore give given Hence hermitian Hilbert space holds identity implies inequality invertible isometric Lemma linear functional linear operator mean metric space non-zero normal normed space Note orthogonal orthonormal particular positive precisely Proof Prove result satisfying separable sequence Show space and let space X spectrum subset Suppose Theorem theory topological space topology uniformly unique unit ball vector space write