Consider the collection of all polynomials (with complex coefficients) of degree <N in x.
(a) Does this set constitute a vector space (with the polynomials as "vectors")? If so, suggest a convenient basis, and give the dimension of the space. If not, which of the defining properties does it lack?
(b) What if we require that the polynomials be even functions?
(c) What if we require that the leading coefficient (i.e. the number multiplying xN−1) be 1?
(d) What if we require that the polynomials have the value 0 at x=1?
(e) What if we require that the polynomials have the value 1 at x=0?