TVS that is of the second category (nonmeager) in must be a dense subset of (since otherwise its closure in would a closed proper vector subspace of and thus of the first category). Sequences of continuous linear maps The following theorem establishes conditions for the pointwise limit of a sequence of continuous linear maps to be itself continuous. If in addition the domain is a Banach space and the codomain is a normed space then Complete metrizable domain proves a weaker form of this theorem with Fréchet spaces rather than the usual Banach spaces. See also Notes Citations Bibliography . . . . . Articles containing proofs Functional analysis Mathematical principles Theorems in functional analysis