Given defined as the space of continuously differentiable functions with inner product .

Let and show:

- is a closed subspace of

I am somewhat uncertain of my solution, thus asking here...

Correct?

Further...

for belongs to thus non-empty.

Assume and . Then . Thus and is closed under addition. Also . Thus and closed under scalar multiplication.

Correct?

But how do I show is a closed subspace? Is it enough to to state it is a continous function defined on a closed and limited interval thus attaining both its minimum and maximum value?