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:
“When hearts have one mingled,
Love first leaves the well-built nest;
The weak one is singled
To endure what it once possessed.
O Love! who bewailest
The frailty of all things here,
Why choose you the frailest,
For your cradle, your home, and your bier.”
—Percy Bysshe Shelley (1792–1822)