The Hilbert Space L2
In Chapter 5, we introduced the general L p (\(\mathbb R\))-spaces. Among the L p (\(\mathbb R\))- spaces, the case p = 2 has a very special status: L 2 \(\mathbb R\) is a Hilbert space, and in fact the only L p (\(\mathbb R\))-space with that property. The space L 2 (\(\mathbb R\))is discussed in Section 6.1. As a continuation and specialization of the previous sections on operators, some fundamental operators on L 2 (\(\mathbb R\)) are considered in Section 6.2. The considered operators will play important roles in the later chapters on the Fourier transform and wavelets. Section 6.3 deals with the Hilbert space L2(a, b); in particular it is shown that the polynomials form a dense subspace of L2(a, b). Section 6.4 discusses Fourier expansions in the framework of the Hilbert space L2(-π, π).
KeywordsHilbert Space Linear Operator Fourier Series Orthonormal Basis Trigonometric Function
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