Thread: Subspaces

for n in W, let T_n = {f in F[x] | deg(f) =n or f =0} and P_n = {f in F[x] | deg(f) <= n or f =0}.
Is T_n a subspace of F[x] over F?
Is P_n a subspace of F[x] over F?

The first is not. consider n=1.

let

so it is not a subspace as it is not closed under addition.

The second is. You can check. Take arbitrary polynomials of degree n or less and add them, they cannot gain degree. Closed under addition

Multiply an arbitrary polynomial of degree n or less by a scalar from F, it will still have degree n or less.