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Published in:
2018 | OriginalPaper | Chapter
12. Hilbert Spaces
Author : Vladimir Kadets
Published in: A Course in Functional Analysis and Measure Theory
Publisher: Springer International Publishing
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Abstract
Among the infinite-dimensional Banach spaces, Hilbert spaces are distinguished by their relative simplicity. In Hilbert spaces we are able to use our geometric intuition to its fullest potential: measuring angles between vectors, applying Pythagoras’ theorem, and using orthogonal projections. Here we do not run into anomalous phenomena such as non-complemented subspaces or, say, linear functionals that do not attain their upper bound on the unit sphere. All separable infinite-dimensional Hilbert spaces are isomorphic to one another. Thanks to this relative simplicity, Hilbert spaces are often used in applications. In fact, whenever possible (true, this is not always the case), one seeks to use the language of Hilbert spaces rather than that of general Banach or topological vector spaces. The theory of operators in Hilbert spaces is developed in much more depth than that in the general case, which is yet another reason why this technique is frequently employed in applications.
