Theorem: Every linear operator
, where
is finite dimensional, is bounded.
Proof 
where
.
What we learn from here is

where
.
Similarly,

where

Another proof of the assertion is

which is a constant.
Note: why does this not work in infinite dimensional spaces? Because the max and min of
and
might not exist.
Like this:
Like Loading...
Related
Published by ayushkhaitan3437
Hello! My name is Ayush Khaitan, and I'm a graduate student in Mathematics.
I am always excited about talking to people about their research. Please please set up a meeting with me if you feel that I might have an interesting perspective to offer- https://calendly.com/ayushkhaitan/meeting-with-ayush
View more posts