Concept

Continuous linear operator

Summary
In functional analysis and related areas of mathematics, a continuous linear operator or continuous linear mapping is a continuous linear transformation between topological vector spaces. An operator between two normed spaces is a bounded linear operator if and only if it is a continuous linear operator. Continuous function (topology) and Discontinuous linear map Bounded operator Suppose that is a linear operator between two topological vector spaces (TVSs). The following are equivalent: is continuous. is continuous at some point is continuous at the origin in If is locally convex then this list may be extended to include: for every continuous seminorm on there exists a continuous seminorm on such that If and are both Hausdorff locally convex spaces then this list may be extended to include: is weakly continuous and its transpose maps equicontinuous subsets of to equicontinuous subsets of If is a sequential space (such as a pseudometrizable space) then this list may be extended to include: is sequentially continuous at some (or equivalently, at every) point of its domain. If is pseudometrizable or metrizable (such as a normed or Banach space) then we may add to this list: is a bounded linear operator (that is, it maps bounded subsets of to bounded subsets of ). If is seminormable space (such as a normed space) then this list may be extended to include: maps some neighborhood of 0 to a bounded subset of If and are both normed or seminormed spaces (with both seminorms denoted by ) then this list may be extended to include: for every there exists some such that If and are Hausdorff locally convex spaces with finite-dimensional then this list may be extended to include: the graph of is closed in Throughout, is a linear map between topological vector spaces (TVSs). Bounded subset Bounded set (topological vector space) The notion of a "bounded set" for a topological vector space is that of being a von Neumann bounded set.