Quick Overview: Some useful results about normed spaces and linear functionals. The axiom of choice in three forms. Cartesian product of a collection of sets. Partially ordered sets and maximal elements. Definition and examples of topological spaces. The subspace topology. Comparison of topologies. Bases for a topology.
Math400 Functional Analysis Section 0 - Detailed Overview & Context
Some useful results about normed spaces and linear functionals. The axiom of choice in three forms. Cartesian product of a collection of sets. Partially ordered sets and maximal elements. Definition and examples of topological spaces. The subspace topology. Comparison of topologies. Bases for a topology. Algebraic bases and dimension. Algebraic complements and quotient spaces. Convexity. Operations on sets (addition and ... Exercises on total boundedness and equicontinuity. Review of normed spaces and linear bounded operators. The dual of a normed space. Examples of dual spaces.
Closed sets, neighborhoods, closure, interior and boundary of a set. Convergence of sequences. Characterization of the closure ... Compactness, relative compactness and, total boundedness. The space C(X,Y) of continuous functions from X to Y. Exercises on the Stone-Weierstrass theorem. Equicontinuity, uniform equicontinuity and pointwise relative compactness. Proof of the Arzela-Ascoli theorem. Dini's theorem. Examples of algebras and subalgebras. The Stone-Weierstrass theorem (without proof). The classical Weierstrass ... Definition and basic properties of the weak star topology of the dual of a normed space.
The box and product topologies. Continuity of projections. Connectedness of a product. The Tychonoff theorem (without proof). Weak compactness and weak sequential compactness. Uniformly convex spaces. Examples and other equivalent ...
