Linear Functional AnalysisThis 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. |
Contents
1 | |
Normed Spaces | 31 |
Inner Product Spaces Hilbert Spaces | 51 |
Linear Operators | 87 |
Duality and the HahnBanach Theorem | 121 |
Linear Operators on Hilbert Spaces | 167 |
Compact Operators | 205 |
Integral and Differential Equations | 235 |
Solutions to Exercises | 265 |
Further Reading | 315 |
Common terms and phrases
adjoint analysis apply arbitrary Banach space bounded called Cauchy Chapter Clearly compact complex Hilbert space condition consider construct contains continuous converges Corollary corresponding defined definition denoted dense discuss eigenvalue element equation equivalent Example Exercise exists extension find finite finite-dimensional first follows formula function give given Hence hold important inequality infinite inner product space integral invertible Lemma Let H linear transformation matrix measure metric space non-zero normed linear spaces normed space notation numbers obtain operator orthogonal projection orthonormal basis particular positive problem proof properties proves respect result Riemann integral roof satisfies self-adjoint separable sequence similar simply solution space and let standard subsequence subset Suppose T G B(H Theorem theory unique vector space write