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I'm currently trying to wrap my head around somebody's (very concise) description of Finite Element Analysis (FEA):

When using FEA, your computed approximation (to the solution of some PDE) will simply be the projection of the exact solution onto your (finite dimensional) basis, in a piecewise (i.e. the domain divided into elements) fashion.

When applying FEA, I just use a set of basis functions (e.g. the Lagrange polynomials), which aren't necessarily orthogonal nor of unit length (they might however form a partition of unity).

For illustrative purposes, let's use the non-orthogonal basis $\{1,x,x^2\}$ for $x \in [0,1]$, in which all quadratic polynomials on $[0,1]$ can be described. While it is quite trivial for this basis, the coefficients for an arbitrary quadratic polynomial in this space can be found using oblique projection (not just simple orthogonal projection since our basis is not orthogonal), as described in this answer.

Actual Question:

For a function that is not in this space of quadratic polynomials on $[0,1]$, for instance $\sin(\pi x)$, how is it projected onto this space (spanned by our non-orthogonal basis $\{1,x,x^2\}$)? Just to be sure, the projection of this function onto our basis is the best approximation possible, right?

So say the solution to something is in fact this function $\sin(\pi x)$, and we want to approximate it with some quadratic polynomial using the mentioned basis (the domain is $[0,1]$). What coefficients will I end up with?

[Edit (1)]: Any references to books that explain FEA from this perspective, or books in which this kind of projection is discussed, are most welcome!

[Edit (2)]: Is the best approximation in this case the same as the $L^2$ projection? An example would be very helpful.

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