The space $c_{00}$ is not a Hilbert space I have to prove that
$$c_{00} = \big\{\: \{x_j\}_{j \in N} \mid x_j = 0 ~\text{eventually} \: \big\} = \operatorname{span}\big\{\:e_j \mid j \in N \:\big\}$$
is not a Hilbert space w.r.t an inner product.
So just to recall the definition of a Hilbert space:
An inner product space is called a Hilbert space if it is complete w.r.t the metric induced by the inner product.
This means, that each Cauchy sequence converges w.r.t the inner product.
So basically what I have to show is that in $c_{00}$ not every Cauchy sequence converges ?
$c_{00}$ is a subspace of $l^2$
 A: (the question was edited after I answered. I'm leaving the more general original answer at the bottom)
As a subspace of $\ell^2(\mathbb N)$, the subspace $c_{00}$ is not complete. This is easily seen by considering the sequence $\{x_n\}$, where
$$
x_n=(1,\frac12,\frac13,\ldots,\frac1n,0,0,\ldots).
$$
Then $\{x_n\}$ Cauchy, but the limit of the sequence is not in $c_{00}$.
Another possibility is to show that $c_{00}$ is dense.

(original answer)
The answer to this depends a bit on how the problem is presented to you.
Usually $c_{00}$ is canonically understood as a subspace of $\ell^\infty(\mathbb N)$, so the norm is
$$
\|x\|_\infty=\max\{|x_n|:\ n\}. 
$$
In this situation is is trivial to check that the infinity norm does not satisfy the parallelogram identity, so it cannot possibly come from an inner product. Indeed, if $\{e_n\}$ denotes the canonical basis,
$$
\|e_1+e_2\|^2+\|e_1-e_2\|^2=1+1=2\ne 4=2\|e_1\|^2+2\|e_2\|^2. 
$$
More substantially, $c_{00}$ cannot be a Banach space, let alone a Hilbert space, in any norm. The reason is that $c_{00}$ has a countable basis; namely, any $x\in c_{00}$ can be written as a linear combination of elements of the canonical basis. Meanwhile, a consequence of Baire's Categoty Theorem is that a Banach space cannot have a countable basis.
