1. ## 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?

2. ## Subspace

The first is not. consider n=1.

let $f(x)=3x -4, g(x)=-3x +2$
$f(x)+g(x)= -2 \not \equiv 0$
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.