According to Wiki a set of elements of a Hilbert space(B) is a basis for that space if:
- Orthogonality: Every two different elements of $B$ are orthogonal: $⟨e_k,e_j⟩=0$ for all $k$, $j$ in $B$ with $k\neq j$.
- Normalization:Every element of the family has norm 1: $⟨e_k,e_k⟩=1$ for all $k$ in $B$.
- Completeness: The linear span of the family $\{e_k \mid k\in B\}$ is dense set in $H$.
Take the Hilbert space to be the $L^2(\mathbb{R})$. Dirac delta functions satisfy the first and second property. Are they densely defined over $B=L^2{(\mathbb{R})}$? It appears to me that it should, since a linear combination of them with the inner product of $L^2$ can make them as close as one wants to a function in $L^2$ because of their point evaluation property. Am I missing something? I couldn't find a reference acknowledging this!
EDIT: As it has been clarified in comments, Dirac deltas are not even functions. I wonder what is then the natural extension of Kronecker deltas(standard basis) to infinite dimensions? (If one sees any finite dimension vector as a discrete function over its index set)