Linear Operator

Let and be Normed Spaces.
A Linear map is an operator if it is continuous.
Invertible Linear Map

Proposition

Let , be normed, Linear.
Then the following are equivalent:

1. is continuous
2. is continuous at
3. is a Bounded Linear Map

Proof

is obvious.

is a nbd of 0 in .
So there is some such that .
Now scale up
Thus for all .

so is Uniformly Continuous.
In particular it is continuous.