Weak follow-up closed set

Weakly sequential set is a term from topology , a sub-area of mathematics . It generalizes the concept of the closed set when it is viewed as the set of all limit values . Sets closed with weak sequences can be found in the discussion of properties of weak topologies and in the solution of distance problems in reflexive Banach spaces .

definition

A standardized space is given . A non-empty subset is called weakly sequence- closed if and only if for every sequence in the set that weakly converges to the weak limit value , the limit value is again in the set . ${\displaystyle (X,\Vert \cdot \Vert )}$ ${\displaystyle M\subset X}$ ${\displaystyle (x_{n})_{n\in \mathbb {N} }}$ ${\displaystyle M}$ ${\displaystyle {\tilde {x}}}$ ${\displaystyle {\tilde {x}}}$ ${\displaystyle M}$

properties

• Every weakly sequence-closed set is also closed, since every convergent sequence also weakly converges. But the reverse is not true.
• Every non-empty closed and convex subset of a normalized space is weakly sequence closed.
• Mazur's theorem follows directly from this : If there is a weakly convergent sequence in a normalized space with a weak limit value , then is . ${\displaystyle (x_{n})_{n\in \mathbb {N} }}$ ${\displaystyle {\tilde {x}}}$ ${\displaystyle {\tilde {x}}\in {\overline {\operatorname {conv} \{x_{n}\,,\,n\in \mathbb {N} \}}}}$
• The epigraph of a function is weakly sequence closed if and only if the function is weakly subsematic .

See also

• Weakly follow 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 .