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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.

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