structural:summarization
Hilbert space theory

The Dirac delta distribution is a densely defined unbounded linear functional on the Hilbert space L of square-integrable functions. Indeed, smooth compactly supported functions are dense in L, and the action of the delta distribution on such functions is well-defined. In many applications, it is possible to identify subspaces of L and to give a stronger topology on which the delta function defines a bounded linear functional.