In mathematics, weak topology is an alternative term for initial topology. The term is most commonly used for the initial topology of a topological vector space (such as a normed vector space) with respect to its continuous dual. The remainder of this article will deal with this case, which is one of the concepts of functional analysis.
One may call subsets of a topological vector space weakly closed (respectively, weakly compact, etc.) if they are closed (respectively, compact, etc.) with respect to the weak topology. Likewise, functions are sometimes called weakly continuous (respectively, weakly differentiable, weakly analytic, etc.) if they are continuous (respectively, differentiable, analytic, etc.) with respect to the weak topology.
Read more about Weak Topology: The Weak and Strong Topologies, The Weak-* Topology, Operator Topologies
Famous quotes containing the word weak:
“We have no participation in Being, because all human nature is ever midway between being born and dying, giving off only a vague image and shadow of itself, and a weak and uncertain opinion. And if you chance to fix your thoughts on trying to grasp its essence, it would be neither more nor less than if your tried to clutch water.”
—Michel de Montaigne (1533–1592)