Vector spaces and projection mappings

Question

Problems:

Vector Spaces and Projection Mappings

Let V be a vector space of all real continuous function on closed interval [ -1, 1].

Let Wo be a set of all odd functions in V and let We bea set of all even functions in V.

(i) Show that Wo and We are subspaces and then show that V= W₀⊕W_e.

(ii) Find a projection mapping onto Wo parallel to We and projection mapping onto We parallel to Wo.

(iii) Let L: V -> V be a mapping that transforms f from V into function that is given by

                L(ƒ)(x):= ∫^x₀ƒ(t) dt. Show that L is linear mapping and state whether the following is true or false:

                L[W₀]⊂W_e and L{W_e]⊂W₀