Introduction to Functional Analysis
The book is written for students of mathematics and physics who have a basic knowledge of analysis and linear algebra. It can be used as a textbook for courses and/or seminars in functional analysis. Starting from metric spaces it proceeds quickly to the central results of the field, including the theorem of HahnBanach. The spaces (p Lp (X,(), C(X)' and Sobolov spaces are introduced. A chapter on spectral theory contains the Riesz theory of compact operators, basic facts on Banach and C*-algebras and the spectral representation for bounded normal and unbounded self-adjoint operators in Hilbert spaces. An introduction to locally convex spaces and their duality theory provides the basis for a comprehensive treatment of Fr--eacute--;chet spaces and their duals. In particular recent results on sequences spaces, linear topological invariants and short exact sequences of Fr--eacute--;chet spaces and the splitting of such sequences are presented. These results are not contained in any other book in this field.
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absolutely convex absolutely convex zero Banach algebra Banach disk Banach space bijection bornological bounded set C*-algebra Cauchy sequence CC(X Chapter choose closed subspace compact subset Consequently continuous linear map converges convex zero neighborhood Corollary countable cr(A Definition denote densely defined DF)-space dual system eigenvalue following hold Frechet space function fundamental system given Hence Hilbert space homomorphism hypothesis implies inductive topology inequality injective isometry isomorphic K-vector space Kothe matrix Lemma linear form linear subspace locally convex space metric linear space metric space normed space nuclear null sequence numbers obtain open subset orthonormal basis orthonormal system pointwise pre-Hilbert space precompact Proof Let property DN Proposition quasi-barrelled reflexive relatively compact Remark representation resp Schwartz space self-adjoint operator seminorms Show spectral measure surjective system of seminorms topological space uniquely determined unitary vector XP(A zero neighborhood basis