Hilbert Space Methods in Science and Engineering

Fundamentals: Linear spaces. Normal spaces. Contractive mappings. Continuous linear operators. The geometry of normed spaces separability. More about complete spaces and compact sets. Finite-dimensional normed spaces. Problems and notes. The geometry of Hilbert spaces: Scalar product. Orthogonal systems (sequences). Some important orthonormal systems in the L2-spaces. The projection principle for finite-dimensional subspace. The projection principle (general case). Some typical examples of the projection principle. Controllability and optimal control of linear systems. Scalar product and bounded linear functionals. Bilinear functionals. First steps in the theory of linear operators on Hilbert spaces. Isomorphic Hilbert spaces and isomophic operators. The conjugate gradient method. Construction of a separating hyperplane. Problems and notes. Reproducing kernel Hilbert space: Hilbert space and kernel. Kernels in the form of an infinite series. A modern approach to the RKHS model. The projection principles in RKHS. Quadrature formulae and splines. Sampling. Conformal mappings and kernels. Gaussian processes. Sobolev spaces and generalised derivative. The finite-element method. Problems and notes. Operator theory: Background from linear algebra. Uniform operator norm and the Neumann series expansion for inverse operators. The spectrum of an operator. Operators with finite-dimensional range. Compact operators. Self-adjoint compact operators. Self-adjoint compact operators. Compact normal operators and the first step towards the representation of non-normal operators. Hilbert-Schmidt operators. Positive operators. Invariant subspaces and projection operators. Non-compact self-adjoint operators. Functional calculus. Problems and notes. Causal operators: Causal operators in L2-spaces. Causality in a Hilbert space. Strictly causal operators. Automatic continuity of causal operators. Appendices. References and further reading. Subject index.