Linear Functional Analysis
This book provides an introduction to the ideas and methods of linear fu- tional analysis at a level appropriate to the ?nal year of an undergraduate course at a British university. The prerequisites for reading it are a standard undergraduate knowledge of linear algebra and real analysis (including the t- ory of metric spaces). Part of the development of functional analysis can be traced to attempts to ?nd a suitable framework in which to discuss di?erential and integral equations. Often, the appropriate setting turned out to be a vector space of real or complex-valued functions de?ned on some set. In general, such a v- tor space is in?nite-dimensional. This leads to di?culties in that, although many of the elementary properties of ?nite-dimensional vector spaces hold in in?nite-dimensional vector spaces, many others do not. For example, in general in?nite-dimensionalvectorspacesthereisnoframeworkinwhichtomakesense of analytic concepts such as convergence and continuity. Nevertheless, on the spaces of most interest to us there is often a norm (which extends the idea of the length of a vector to a somewhat more abstract setting). Since a norm on a vector space gives rise to a metric on the space, it is now possible to do analysis in the space. As real or complex-valued functions are often called functionals, the term functional analysis came to be used for this topic. We now brie?y outline the contents of the book.
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a e H adjoint algebraic arbitrary Banach space bounded Cauchy sequence Cauchy–Schwarz inequality characteristic value closed linear subspace compact operator complex Hilbert space continuous functions converges convex Corollary 3.36 countable decomposition definition denote dense dimensional eigenvalue eigenvector element en)en equivalent Example Exercise finite rank finite-dimensional follows immediately formula function f Hahn–Banach theorem Hence Hilbert–Schmidt inner product space integral equation integral operator invertible isometrically isomorphic isometry Lebesgue integration Lemma let f Let H let Te B(H linear functional linear operator linear transformation LP(X matrix metric space non-empty non-zero eigenvalues normed linear spaces normed space normed vector space notation obtain orthogonal projection orthonormal basis orthonormal sequence polynomial Proof Let properties real numbers reflexive satisfies separable space and let space H spectrum subset Suppose that H unitary