Functional Analysis 1

The main purpose of the course is to introduce students to the basic concepts and theorems of Functional Analysis, and to develop an “abstract geometrical” approach to different mathematical problems.

1. Metric spaces. Properties of complete metric spaces. Mappings of metric spaces. Fixed points. Application to ODE. Compact sets in metric spaces.
2. Normed spaces. Holder’s inequality. The spaces: $C[0,1]$, $l_p$, $c_0$, $c$, $L_p$. Banach spaces. The completion theorem. Completeness of the above spaces.
3. Linear operators. The norm of a linear operator. Isomorphism and isometry. Any two $n$-dimensional normed spaces are isomorphic. The open mapping theorem and the closed graph theorem. Uniform boundedness principle. The completeness of the space $L(X, Y)$ of all linear bounded operators acting from a normed space $X$ into a Banach space $Y$.
4. Linear functionals. The Hahn–Banach extension theorem. The dual space. Dual spaces for the examples above. Adjoint operators.
5. Quotient space. Completeness of the quotient space. The dual space for a quotient space. Reflexive spaces.
6. Hilbert spaces. The Cauchy–Schwarz inequality. Orthogonal projectors. Orthonormal basis. Any two separable infinite-dimensional Hilbert spaces are isometric. Closed operators in a Hilbert space.