Kreysig's Functional Analysis 3.2 Question 3 Let $X$ be the inner product space consisting of the polynomial $x=0$ and all real polynomials in $t$, of degree, not exceeding 2. considered for real $t\in[a,b]$, with inner product defined by $$\langle x,y\rangle=\int_{a}^{b}x(t)y(t)dt.$$
Show that $X$ is complete. 
My proof is as followed. 
Since $X$ is an inner product space, by the completion of inner product space theorem, there exists a Hilbert space $H$ and an isomorphism $A$ from $X$ onto a dense subspace $W\subset H$. Thus, $X\cong W\subset H$. Since $X$ is a finite-dimensional with $dim(X)=3$ and $X\subset H$, where $H$ is a Hilbert space, it follows that $X$ is complete. 
Is my proof correct?
 A: A finite dimensional normed space is complete. Your space $X$ has dimension $3$, the norm on $X$ is $||x||= \langle x,y\rangle ^{1/2}.$
A: The non-trivial part is the following:
If $X$ is a finite dimensional normed space, then $X$ is complete.
However, it is much easier to prove that  
If $X$ is a finite dimensional inner product space, then $X$ is complete.
Proof. Let $\dim X=N$ and $\{e_1,\ldots, e_N\}$ an orthonormal basis of $X$. Assume now that $\{x_n\}\subset X$ is a Cauchy sequence. Then, for every $k=1,\ldots,N$, the sequence $\{\langle x_n, e_k\rangle\}\subset \mathbb R$ is a Cauchy sequence, and hence convergent. Say $\langle x_n, e_k\rangle\to c_k$ and set
$$
x=\sum_{k=1}^N c_kx_k.
$$
Then
$$
\|x_n-x\|^2=\left\|\sum_{k=1}^N \big(\langle x_n, e_k\rangle-c_k\big)e_k\,\right\|^2=\sum_{k=1}^N(\langle x_n, e_k\rangle-c_k)^2\to 0.
$$
A: Let $(e_{2},e_{1},e_{0})$ be an orthonormal basis for $X$ with respect to that inner product, given a Cauchy sequence $(x_{n})$ in $X$, write $x_{n}a_{n}e_{2}+b_{n}e_{1}+c_{n}e_{0}$, then 
\begin{align*}
\|x_{n}-x_{m}\|^{2}=|a_{n}-a_{m}|^{2}+|b_{n}-b_{m}|^{2}+|c_{n}-c_{m}|^{2}.
\end{align*}
So the sequences $(a_{n}), (b_{n}), (c_{n})$ are all Cauchy and hence convergent, say, $a_{n}\rightarrow a, b_{n}\rightarrow b, c_{n}\rightarrow c$. Now we see that with $x=ae_{2}+be_{1}+ce_{0}$
\begin{align*}
\|x_{n}-x\|^{2}=|a_{n}-a|^{2}+|b_{n}-b|^{2}+|c_{n}-c|^{2}\rightarrow 0
\end{align*}
as $n\rightarrow\infty$.
