Linear Functional AnalysisThis book provides an introduction to the ideas and methods of linear func tional analysis at a level appropriate to the final 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 the ory of metric spaces). Part of the development of functional analysis can be traced to attempts to find a suitable framework in which to discuss differential and integral equa tions. Often, the appropriate setting turned out to be a vector space of real or complex-valued functions defined on some set. In general, such a vector space is infinite-dimensional. This leads to difficulties in that, although many of the elementary properties of finite-dimensional vector spaces hold in infinite dimensional vector spaces, many others do not. For example, in general infinite dimensional vector spaces there is no framework in which to make sense of an alytic 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 briefly outline the contents of the book. |
Contents
Normed Spaces | 31 |
Inner Product Spaces Hilbert Spaces | 51 |
1 | 87 |
5 | 123 |
Compact Operators | 161 |
Integral and Differential Equations | 191 |
Solutions to Exercises | 221 |
Further Reading | 265 |
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adjoint Banach space basis for H bounded linear Cauchy sequence Chapter characteristic value closed linear subspace compact operator complex Hilbert space consider construct continuous functions Corollary corresponding countable d₁ Definition denoted dense differential equations eigenfunctions eigenvalue eigenvector elements en)en equivalent Example Exercise finite rank finite-dimensional spaces formula Fredholm function ƒ functional analysis Hence Hilbert-Schmidt infinite inner product space integral equation integral operator invertible isometry kernel L¹(X Lebesgue integration Lemma Let H lim xn linear operator linear subspace linear transformation matrix metric space non-zero eigenvalues normed linear spaces normed space normed vector space notation orthogonal projection orthonormal basis orthonormal sequence polynomial positive properties proves real numbers result scalar space and let space H space over F spectrum square root standard norm subset Suppose theory unitary zero