Optimization—Theory and Applications pp 325-366 | Cite as

# Closure and Lower Closure Theorems under Weak Convergence

Chapter

## Abstract

If *X* is a normed linear space over the reals with norm ‖*x*‖, let *X** be the dual of *X*, that is, the space of all linear bounded operators *x** on *X*, the linear operation being denoted by (*x**, *x*), or *X** × *X* → *R*. A sequence [*x*_{ k }] of elements of *X* then is said to be convergent in *X* to *x* provided ‖*x*_{ k } – *x*‖ → 0 as *k* → *∞*. A sequence [*x*_{ k }] of elements of *X* is said to be weakly convergent in *X* to *x* provided (*x**, *x*_{ k }*)* → (*x**, *x*) as *k* → ∞ for all *x** *∈* *X**.

## Keywords

Weak Convergence Lower Semicontinuous Measure Zero Measurable Subset Equivalence Theorem
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## Copyright information

© Springer-Verlag New York Inc. 1983