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:
“Why, therefore, should we do ourselves this wrong,
Or others—that we are not always strong—
That we are sometimes overborne with care—
That we should ever weak or heartless be,
Anxious or troubled—when with us is prayer,
And joy and strength and courage are with Thee?”
—Richard Chenevix Trench (1807–1886)