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:
“A country is strong which consists of wealthy families, every member of whom is interested in defending a common treasure; it is weak when composed of scattered individuals, to whom it matters little whether they obey seven or one, a Russian or a Corsican, so long as each keeps his own plot of land, blind in their wretched egotism, to the fact that the day is coming when this too will be torn from them.”
—Honoré De Balzac (1799–1850)