Suppose we let $$L^{p=2}(D,\mathbb{R})$$ denote a set of real functions on a domain D such that if $$\mathbf{a} \in L^{p=2}(D,\mathbb{R})$$ then we have $$\int_{D} \left | a(t) \right |^{2}dt<\infty$$
This is from my notes which leaves me very confused.
In the general case, the $$L^{p}-space$$ is defined as
$$\left \| f \right \|_{p}=(\int_{D}\left | f \right |^{p}d\mu )^{\frac{1}{p}}$$
In the case where p=2, the expression outlined in my notes appears to be suspect.
Well, these notes are very brief. In reality, one first defines the calligraphic L $^p$ space consisting of Lebesgue measurable functions which satisfy $\|f\|_p=(\int\limits_{D}{|f|^p})^{1/p}<\infty$. Now this is still not a norm, because not all functions with this quantity being zero are identically zero functions - they can be nonzero on sets with measure 0. Therefore this space is partitioned into equivalence classes with respect to the equivalence relation "equal almost every where", i.e $f$ and $g$ are in one and the same equivalence class iff $f-g=0$ a.e, which is iff $\|f-g\|_p=0$. Then the resulting space of equivalence classes is denoted by $L^p$ (not calligraphic) and is normed using the same expression $\|f\|_p$ which now becomes a real norm, because only the zero element has a norm zero. To understand this better, I suggest that you read about equivalence relations and partitioning, as well as about seminorms. Maybe the book "Measure theory and integration" by Michael E. Taylor, Chapter 4 will help you. The construction of $L^p$ is informally summarized there.