Continuous linear operator
If for every there exists a such that we say the operator between normed spaces is continuous.
A continuous linear operator maps bounded sets into bounded sets. A linear functional is continuous if and only if its kernel is closed. Every linear function on a finite-dimensional space is continuous.
The following are equivalent: given a linear operator A between topological spaces X and Y:
- A is continuous at 0 in X.
- A is continuous at some point in X.
- A is continuous everywhere in X.
The proof uses the facts that the translation of an open set in a linear topological space is again an open set, and the equality
for any set D in Y and any x0 in X, which is true due to the additivity of A.