Math400 Functional Analysis Section 2

The box and product topologies. Continuity of projections. Connectedness of a product. The Tychonoff theorem (without proof).

Closed sets, neighborhoods, closure, interior and boundary of a set. Convergence of sequences. Characterization of the closure ...

We prove some basic facts about hyperplanes and we formulate the problem of separating two convex disjoint subsets of a ...

Definition and examples of topological spaces. The subspace topology. Comparison of topologies. Bases for a topology.

The weak and strong topologies of a normed space coincide if and only if the space is finite dimensional. In an infinite ...

We prove the two geometric forms of the Hahn-Banach theorem and we formulate a method for proving that a subspace is dense.

Equicontinuity, uniform equicontinuity and pointwise relative compactness. Proof of the Arzela-Ascoli theorem.

Proof of the Hahn-Banach theorem.

Definition and examples of subnorms. The analytic form of the Hahn- Banach theorem and three of its corollaries.

Reflexivity, separabilty and duals of Lp spaces.

We define the bidual of a normed space E and estabish an isometry from E to its bidual. We give the definition of a reflexive space ...

Review of normed spaces and linear bounded operators. The dual of a normed space. Examples of dual spaces.

Dini's theorem. Examples of algebras and subalgebras. The Stone-Weierstrass theorem (without proof). The classical Weierstrass ...

A smaller topology contains more compact sets, more connected sets, more convergent sequences but fewer real valued ...

Convex sets and linear operators in the weak topology. The weak topology as a vector space topology.