a) Is true because is a continuous linear functional on . If we denote by this continuous mapping the space is

.

b) Is false. To prove it just observe that there are dense subspaces in formed by integrable functions and with integral 0. Look at the Haar system Haar wavelet - Wikipedia, the free encyclopedia.

If the set of functions with integral 0 were closed, this would imply that the closure of the span of the Haar system, that is the whole , would be included in , that certainly is not true.