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:
“It is useless to check the vain dunce who has caught the mania of scribbling, whether prose or poetry, canzonets or criticisms,—let such a one go on till the disease exhausts itself. Opposition like water, thrown on burning oil, but increases the evil, because a person of weak judgment will seldom listen to reason, but become obstinate under reproof.”
—Sarah Josepha Buell Hale 1788–1879, U.S. novelist, poet and women’s magazine editor. American Ladies Magazine, pp. 36-40 (December 1828)