Hilbert Spaces

A Hilbert space is a special kind of Banach space, where the norm is induced by the inner product. All of the results for Banach spaces are valid in Hilbert spaces. The special feature of Hilbert space is that the Projection Theorem 13.13 holds. The dual of a Hilbert space is easily characterized in Riesz-Fréchet Theorem 13.15, and a Hilbert space is reflexive. There exists a norm preserving conjugate linear map from the Hilbert space to it dual. We define two vectors in the Hilbert space to be orthogonal using this map. Then, Hermitian adjoint of a bounded linear operator between Hilbert spaces is presented together with its properties. In Sect. 13.5, I present the Gram–Schmidt procedure and other results related to orthonormal sequences in Hilbert spaces. The Legendre polynomials is presented in Example 13.34. The L2 theory for Fourier series is presented in Example 13.36. The Projection Theorem is revisited in Sect. 13.6, and I present the minimum norm problem for convex sets. Then, I define positive-definite and positive semi-definite operators among Hermitian operators on a Hilbert space, and derive their properties in Sect. 13.7. Section 13.8 is devoted to the study of projection operators on Hilbert space, and the pseudoinverse of bounded linear operators between Hilbert spaces. The last section of this chapter presents the study of spectral theory for compact bounded linear operators between Hilbert spaces. In Example 13.57, I establish that first order Sobolev space W 2 , 1 , x 0 ( 𝕏 , Y ) $$W_{2,1,x_0}(\mathbb {X},\mathscr {Y})$$ is a Hilbert space when 𝕏 $$\mathbb {X}$$ is a σ-finite metric measure subspace of ℝ m $$\mathbb {R}^m$$ and Y $$\mathscr {Y}$$ is separable Hilbert space with Y ∗ $$\mathscr {Y}^*$$ being separable.