Consider the Hilbert Space of Lebesgue integrable real valued function on with inner product .
Prove that the set of all even function in is a closed subspace of .
Well, you know that all you have to prove is that a non-empty subset of vector space is closed under addition and scalar multiplication to prove it is a subspace, don't you? So you want to prove that the sum of two even functions is an even function and that the product of a number and an even function is an even function.
Well, proving it's a subspace is quite easy. For the fact that it's closed you'll need the following:
1) If is a sequence of integrable functions such that then there exist a subsequence such that a.e. on
Take such that then and so a.e. on .
2)If is a sequence of integrable functions such that in then there exists a subsequence such that a.e. on .
Since by (1) we have that there exists a subsequence such that a.e. on and so the result follows. Notice that but in fact the case is easier since you don't even need to pick a subsequence (you see why?).
Now let be even functions such that in then there exists a subsequence such that a.e. on and so let then it's obvious and if then . Now just define if and otherwise then in and is even.