Abstract
This chapter covers the necessary concepts from linear functional analysis on Hilbert and Banach spaces: in particular, we review here basic constructions such as orthogonality, direct sums and tensor products.
Dr. von Neumann, ich möchte gern wissen, was ist dann eigentlich ein Hilbertscher Raum?
David Hilbert
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Notes
- 1.
To be more precise, as with the Lebesgue L p spaces, Sobolev spaces consist of equivalence classes of such functions, with equivalence being equality almost everywhere.
- 2.
Or even just a topological vector space.
- 3.
Completions of normed spaces are formed in the same way as the completion of \(\mathbb{Q}\) to form \(\mathbb{R}\): the completion is the space of equivalence classes of Cauchy sequences, with sequences whose difference tends to zero in norm being regarded as equivalent.
- 4.
Of course, Fourier did not use the modern notation of Hilbert spaces! Furthermore, if he had, then it would have been ‘obvious’ that his claim could only hold true for L 2 functions and in the L 2 sense, not pointwise for arbitrary functions.
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Sullivan, T.J. (2015). Banach and Hilbert Spaces. In: Introduction to Uncertainty Quantification. Texts in Applied Mathematics, vol 63. Springer, Cham. https://doi.org/10.1007/978-3-319-23395-6_3
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DOI: https://doi.org/10.1007/978-3-319-23395-6_3
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