These spaces are inner product spaces with a complete L2-like norm metric and each of these is consequently a Banach space, though the converse does not hold.
Generalizations of these entities that include representations of Schwartz distributions are sometimes known as Gelfond doubles, but are more generally denoted by prefixing their usual name with "rigged." Stability for dynamic systems on these objects is the subject of the von Neumann ergodic theorem. One class of operators over these forms a self-adjoint ideal for the associated Banach algebras; those operators are named for Schmidt and for the namesake of these and are bounded in the namesake norm that defines these spaces. These spaces are inner product spaces with a complete L2-like norm metric and each of these is consequently a Banach space, though the converse does not hold. For ten points, identify these vector spaces in which the squared length of a vector is the inner product of the vector with itself and which are named for a German mathematician.
Generalizations of these entities that include representations of Schwartz distributions are sometimes known as Gelfond doubles, but are more generally denoted by prefixing their usual name with "rigged.
{Hilbert} spaces