# Extend the functional by continuity (Functional analysis)

1. Mar 14, 2012

### mathdunce

1. The problem statement, all variables and given/known data

Let E be a dense linear subspace of a normed vector space X, and let Y be a
Banach space. Suppose T0 $\in$ £(E, Y) is a bounded linear operator from E to Y.
Show that T0 can be extended to T$\in$ £(E, Y) (by continuity) without increasing its norm.

3. The attempt at a solution
Someone kindly gave me a hint. I am trying to work out the details. The deadline is approaching. So I put the question here just in case. Thanks.
For this particular problem you want to show that if (xn) converges to x then T0(xn) is a Cauchy sequence and then define f(x) as the limit of the sequence. Finally you need to show that the map is a well-defined bounded linear function.

2. Mar 14, 2012

### micromass

Staff Emeritus
Use that T0 is uniform continuous.

3. Mar 14, 2012

Thank you!