The dual space of is (isometrically isomorphic to) . If and then the value of the functional y at the point x is . This is a very standard result which you should be able to find in any relevant textbook.

This is more interesting. Take the second question first. If g(x) is in the orthogonal complement of Y, then for all f in Y. But . If you change variables in the two integrals, putting y=1-x in the first and y=1+x in the second, then you find that (remembering that f(1+y)=f(1-y)). This holds for all continuous functions on [0,1], so you conclude that g(1+y)=–g(1-y) for all y in [0,1].

Thus the orthogonal complement of Y is If you repeat the same argument on the subspace Z, you will find that its orthogonal complement is Y. Therefore Y is equal to its second orthogonal complement, which is closed in X. Hence Y is closed.

This is another standard piece of bookwork. Look it up in any good textbook. The proof uses Hölder's inequality.