How could one find a nontrivial example (a subspace which contains more than simply the zero vector) of an infinite dimensional vector space that contains a finite dimensional subspace and prove it's truly a subspace and has finite dimension?
What comes to mind is the (infinite) set of polynomials $P_{\infty}$, and some subspace of $P_n$ of degree and dimension $n$.
Though it seems fairly obvious why $P_n$ is truly a subset of $P$, how could this be said more concretely and analytically?
I figured that since $P_{\infty} = \{c_{0}x^{0}, c_{1}x^{1}, c_{2}x^{2},...,c_{n}x^{n},...|c_1,c_2,c_3,...,c_n,...\in\mathbb{R}\}$, then $P_{n}$ is seen to be contained in $P_{\infty}$ as $P_{n}=\{c_{0}x^{0}, c_{1}x^{1}, c_{2}x^{2},...,c_{n}x^{n}|c_1,c_2,c_3,...,c_n\in\mathbb{R}\}$.
Then to prove it is truly a subspace the subspace theorem could be applied.
However, I am uncertain of the best approach to finding the dimension of this subspace (or any for that matter).
Should I just determine a basis for the space and then use the knowledge that the dimension of a space whose basis $B$ has $n$ vectors is equal to $n$?