If A Linear Map Is Finite Dimensional, The Linear Map Is Continuous
Let X and Y be two normed spaces and f a linear map from X to Y. If X is finite-dimensional, choose a base (e1, e2, …, en) in X which may be taken to be unit vectors. Then,
and so by the triangle inequality,
Letting
and using the fact that
for some C>0 which follows from the fact that any two norms on a finite-dimensional space are equivalent, one finds
Thus, f is a bounded linear operator and so is continuous.
If X is infinite-dimensional, this proof will fail as there is no guarantee that the supremum M exists. If Y is the zero space {0}, the only map between X and Y is the zero map which is trivially continuous. In all other cases, when X is infinite dimensional and Y is not the zero space, one can find a discontinuous map from X to Y.
Read more about this topic: Discontinuous Linear Map
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