Mathematics

1) Ordered sets: binary relations, order, majorant, minority, ascending or descending filtering orders, lattice, total order, good ordering, examples; axiom of choice, principle of good ordering, Zorn's lemma; equivalence between AC, Zorn, principle of good order (sd); topology, recalls on the axioms of numerability and relations. Weakest and strongest topology among a family of topologies given on the same set; separability; net and ordinary successions. 2) net, subnet, accumulation point for a net; universal nets and convergence at the points of accumulation; existence of universal nets; 3) Filter, universal net, existence of universal subnets. Basic properties of continuous functions on compacts. Compactness and equivalent definitions; elementary properties; weak topology; product topology. Tychonoff's theorem. Comments on cases in which AC is not necessary, finite or countable set of indices. Property T2. Product of spaces T2 is and T2. Compact T2 is T4. Countable product of compact metrics is compact metric in the product topology. Tychonoff's cube. 4) The subsets of Tychonoff's cube are normal metric spaces with a numerable basis. Theorem (sd) Any topological space normal with a countable basis and 'homeomorphic to a subset of the Tychonoff cobo, and therefore metrizable. It is compact if and only and the subset is closed. Relations between topology and net. Normed spaces; Banach spaces, basic properties; linear operator between normed spaces, equivalent conditions of continuity. Space of linear and bounded operators. Sub-multiplication of the standard. Quotient normed space. Injective operator associated with a bounded and linear operator. Banach space quotient. If X is normed, M and X / M are Banach's, then X is Banach's. 5) A subspace of finite dim in a normed space is Banach. Extension of a densely defined operator, recalls on the completion of a metric space., Realization of the completion for a normed space as a canonical immersion in the bidual. Various examples of normed spaces and completion of infinite dimension, continuous functions on a locally compact Hausdorff space, examples with infinite products. Baire's category theorem. 6) Open application theorem, continuity of the inverse of a limited and bi-univocal operator between Banach spaces; closed graph theorem; principle of uniform limitation. 7) Application to the space of continuous and periodic functions in an interval; Fourier algebra on the circle; exercise on the existence of non-derivable functions at any point of an interval. Minkowski functional, extensibility of functionals; theorem of Hahn-Banach, existence of linear and bounded functionals on a normed space that separate the points. Extension to the quotient space. Subspace annihilator. Canonical isometry of immersion in the bidual and relationship with that of completion. Relation to reflexivity. 8) Classical examples of reflexive Banach spaces. Non-degenerate duality. Added operator. Algebraic properties. Isometry of the adjunct. Automatic boundedness of a linear operator with addition between Banach spaces. Exercises on classical examples of duality between spaces of sequences. Weak topology defined by a family of seminorms; topological vector space; locally convex vector space. Continuous functionals with respect to the topology defined by a duality. Construction of Minkowski functionals in a topological vector space. Equivalence between the two definitions of SVTLC. 9) Existence of continuous functionals that separate the convex ones in an SVT (Geometric form of the Hahn-Banach theorem). Weak topology of a normed space, equality of the dual with respect to the original topology; the two topologies have the same closed convex. ∗ -weak topology of the dual, computation of the topological dual. Alaoglu's theorem. 10) Sequential Alaoglu's theorem, independence from the axiom of choice; consequences of Alaoglu's theorem: the unitary disk closed in norm of a Banach space is weakly compacted if and only if the space is reflexive (only half proof). Existence of weakly convergent ordinary subsequences of normally bounded ordinary sequences in a reflexive Banach space (Eberlein-Smulian, s.d.), example with Lp (X) with 1

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