1 Hilbert spaces -- 1.1 Complete sets, Fourier expansions -- 1.1.1 Preliminary notions. Subspaces. Complete sets -- 1.1.2 Fourier expansions -- 1.1.3 Harmonic functions; Dirichlet and Neumann Problems -- 1.2 Linear operators -- 1.2.1 Linear operators defined giving T en = vn, and related Problems -- 1.2.2 Operators of the form T x = v(w;x) and T x = ån vn(wn;x) -- 1.2.3 Operators of the form T f (x) = j(x) f (x) -- 1.2.4 Problems involving differential operators -- 1.2.5 Functionals -- 1.2.6 Time evolution Problems. Heat equation -- 1.2.7 Miscellaneous Problems -- 2 Functions of a complex variable -- 2.1 Basic properties of analytic functions -- 2.2 Evaluation of integrals by complex variable methods -- 2.3 Harmonic functions and conformal mappings -- 3 Fourier and Laplace transforms. Distributions -- 3.1 Fourier transform in L1(R) and L2(R) -- 3.1.1 Basic properties and applications -- 3.1.2 Fourier transform and linear operators in L2(R) -- 3.2 Tempered distributions and Fourier transforms -- 3.2.1 General properties -- 3.2.2 Fourier transform, distributions and linear operators -- 3.2.3 Applications to ODE's and related Green functions -- 3.2.4 Applications to general linear systems and Green functions -- 3.2.5 Applications to PDE's -- 3.3 Laplace transforms -- vvi Contents -- Groups, Lie algebras, symmetries in physics -- 4.1 Basic properties of groups and representations -- 4.2 Lie groups and algebras -- 4.3 The groups SO3; SU2; SU3 -- 4.4 Other direct applications of symmetries to physics -- Answers and Solutions. .