Linear Algebra – Formal Power Series

Question 1: Let x be a variable. Define a formal power series in the variable x over a field F to be a sum of the form

a_0 + (a_1)(x) + (a_2)(x^2) + (a_3)(x^3) + … = SUM (t=0, infinity) (a_t)(x^t)

with a_t is a real number of F. Let F[[x]] be the set of all formal powers series in x over F. Define an addition on F[[x]] by adding the coefficients of like powers of x, and define a scalar multiplication (with scalars in F) by multiplying all coefficients by the scalar. In other words (see attached).

Prove that F[[x]] is a vector space over F with the given addition and scalar multiplication

Question 2: Let P(F) be the set of all polynomials over a field F. For each of the following, determine (with proof) if the subset is a subspace of P(F).

i) (P_m)(F) = {p is an element P(F) | deg(p) <= m}
ii) U = {p is an element of P(F) | deg(p) = 4}
iii) W = {p is an element of P(F) | p(1) = 0}
iv) W = {p is an element of P(F) | p(1) =/ 0}

Question 3: Suppose that U and W are finitely-generated subspaces of a vector space V. Prove that U + W is also finitely-generated by finding an explicit spanning set of U + W.