you need to define what kind of function p(t) is, which vector space we're taking as our "parent" space, for the first 3 sets.

i will address number 1:

we check for the 3 conditions:

a) the 0-function (which is constant for all x in [a,b]) has the property that 0(a) = 0(b), since both of these are 0, so it is in S.

b) suppose f,g are in S. we want to show that f+g is in S. now, since f and g are in S, f(a) = f(b), g(a) = g(b). therefore:

(f+g)(a) = f(a) + g(a) = f(b) + g(b) = (f+g)(b), so f+g is in S.

c) let k be a real number, and left f be any element of S. then (kf)(a) = k(f(a)) = k(f(b)) = (kf)(b), so kf is in S.

so S is indeed a subspace of V.