Definitions for L2 and Lp Spaces? I am taking a course in Functional Analysis online, and unfortunately some important terms have not been well defined.  In particular, isn't L2 space just Lp space with p=2 ?  If so, why aren't continuous functions on closed intervals with the L2 norm Banach spaces (on finite dimensional spaces, by the Fischer-Riesz Theorem)? 
 A: If you ignore the inner product, $L^2$ space is just $L^p$ space with $p=2$. The inner product gives $L^2$ additional structure: it is not only a Banach space but also a Hilbert space.
The subset of continuous functions in $L^p$ (for $1 \leq p < \infty$) does not form a Banach space because it is not complete. However, it is a dense subset of $L^p$, meaning that given any $f \in L^p$, there is a sequence $f_n$ of continuous functions converging to $f$ in the $L^p$ norm. Putting it another way, an arbitrary $f \in L^p$ can be approximated arbitrarily closely (with respect to the $L^p$ norm) by continuous functions.

Regarding the definitions: A Banach space is by definition a complete normed vector space. $L^2$ is a complete normed vector space which also has an inner product. So it is both a Banach space and a Hilbert space. For any $p<\infty$ (including $p=2$), the subset of $L^p$ consisting of the continuous functions is a normed vector space, but it is not complete, so it is not a Banach space.
