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?