Representation of linear functionals I have always seen the linear functionals in $R^n$ expressed at $\ell(x) = \sum_{i=0}^n a_ix_i$ And in an countable metric space $\ell(x) = \sum_{i=0}^{\infty} a_ix_i$. I guess that this follows directly from http://en.wikipedia.org/wiki/Riesz_representation_theorem, for Hilbert spaces.
But what if we are not in an Hilbert space or if the space is uncountable.
If X was a 1 dimensional space I would get $f(x) = f(1)x$ by continuity and linearity (by derivation and integration) and by partial derivation it would look like $f(x) = \sum_{i=0}^n f(1)x_i$
for the n dimensional case
 A: In general, linear functionals can be very different than that. 
The easiest example of a functional not of the form you claim, is probably there case where you take $X=C[0,1]$ and consider functionals like $\ell(f)=f(0)$ (or any other point, for that matter). 
Note also that when you consider infinite-dimensional spaces, they most often come with a topology, and one considers bounded (i.e. continuous)  linear functionals. Unbounded linear functionals exist but are a lot tricker to deal with. 
A: I can think of a nice representation theorem that holds in a non-Hilbert space.  It goes by the name Riesz-Kakutani-Markov:
Let $X$ be a compact Hausdorff space and $(C(X),\|\cdot\|_\infty)$ the space of continuous real valued functions on $X$ endowed with the maximum norm.  Then, every bounded linear functional $F$ on $C(X)$ can be written as an integral against a signed, finite Borel measure $\mu$ on $X$:
$$
F(f)=\int_X fd\mu
$$
with norm
$$
\|F\|=\int_X\vert d\mu\vert
$$ where $\vert d\mu\vert$ is the absolute variation of $\mu$.
A good resource for this theorem is Lax: Functional Analysis.  Granted, this is more sophisticated than the Riesz representation theorem on Hilbert spaces, but that's to be expected.
