• Review on the locally convex spaces and examples: C C(Ω), Ck(Ω),DK(Ω)
• Definition of a distribution and preliminary properties (weak topology on the space of distributions).
Leibnitz Formula and Study of almost regular cases, Stokes and Green's formula and its application for finding derivations: Fundamental solution of Laplace.
• Tensor Product. Fubini's Theorem and density Theorem.
• Complement on topology: Banach-Steinhaus theorem and Frechet spaces (application to functional spaces).
• Topology of D(Ω): the space of distributions is the topological dual of D(Ω). More properties in D’(Ω).
• Convolution of distributions.
• Fourier transform (L1(Ω), the space S, tempered distributions S’, L2(Ω)).
• Fourier transformation of distributions. Fourier transformation and convolution: application for solving a linear PDE.