Linear Functional Analysis (Google eBook)

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Springer Science & Business Media, Dec 29, 2007 - Mathematics - 324 pages
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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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Contents

I
1
II
2
III
11
IV
20
V
31
VI
39
VII
45
VIII
51
XIX
132
XX
137
XXI
144
XXII
155
XXIII
159
XXIV
166
XXV
176
XXVI
183

IX
60
X
65
XI
72
XII
82
XIII
87
XIV
96
XV
104
XVI
108
XVII
121
XVIII
127
XXVII
192
XXVIII
205
XXIX
253
XXX
264
XXXI
315
XXXII
317
XXXIII
318
XXXIV
319
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