Weakly follow compact set

The concept of the weakly sequence-compact set and the weakly * sequence-compact set is a term from topology , a sub-area of mathematics . It is a generalization of the compactness of the sequence for topologies that are coarser than the standard topology, the so-called weak topology and the weak - * - topology . Weakly sequence compact sets are important for the fundamentals of mathematical optimization , since a certain class of functions assumes a minimum on weak sequence compact sets and thus guarantees the solvability of optimization problems.

definition

A standardized space is given . A nonempty subset is called weakly sequence- compact if every sequence in this set has a weakly convergent subsequence whose weak limit value belongs to again . ${\displaystyle (X,\Vert \cdot \Vert )}$ ${\displaystyle M\subset X}$ ${\displaystyle M}$

If the dual space of , then a set is said to be weak * sequence- compact if every sequence in this set has a weak * convergent subsequence whose weak * limit value belongs to again . ${\displaystyle X'}$ ${\displaystyle X}$ ${\displaystyle M'\subset X'}$ ${\displaystyle M'}$

properties

• If the normalized space is finite dimensional, then the set is weakly sequential compact if and only if it is closed and bounded. ${\displaystyle M}$
• According to Eberlein – Šmulian's theorem , weak sequence compactness and weak compactness coincide for weakly closed sets in Banach spaces.
• If separable, then every closed sphere is sequentially compact in weak *. ${\displaystyle X}$ ${\displaystyle X'}$
• If a reflexive Banach space, then every closed sphere is weakly sequential compact. ${\displaystyle X}$

use

In addition to the discussion of weak topologies, sets with weak consequences also appear in the optimization . Here they provide statements about the existence of extreme locations. Weakly sub-continuous functions always assume a minimum on a weakly sequential set.

See also

• Weak relatively compact set

literature

• Johannes Jahn: Introduction to the Theory of Nonlinear Optimization . 3. Edition. Springer-Verlag, Berlin Heidelberg New York 2007, ISBN 978-3-540-49378-5 .
• Hans Wilhelm Alt: Linear Functional Analysis . 6th edition. Springer-Verlag, Berlin Heidelberg 2012, ISBN 978-3-642-22260-3 , doi : 10.1007 / 978-3-642-22261-0 .