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.