is a commutative C*-algebra with pointwise multiplication and conjugation. For many measure spaces, including all sigma-finite ones, it is in fact a commutative von Neumann algebra. An element of defines a bounded operator on any space by multiplication. For the spaces are a special case of spaces, when consists of the natural numbers and is the counting measure on More generally, if one considers any set with the counting measure, the resulting space is denoted For example, the space is the space of all sequences indexed by the integers, and when defining the -norm on such a space, one sums over all the integers. The space where is the set with elements, is with its -norm as defined above. As any Hilbert space, every space is linearly isometric to a suitable where the cardinality of the set is the cardinality of an arbitrary Hilbertian basis for this particular Properties of Lp spaces As in the discrete case, if there exists such that then Dual spaces The dual space (the Banach space of all continuous linear functionals) of for has a natural isomorphism with where is such that (i.e. ). This isomorphism associates with the functional defined by for every The fact that is well defined and continuous follows from Hölder's inequality. is a linear mapping which is an isometry by the extremal case of Hölder's inequality. It is also possible to show (for example with the Radon–Nikodym theorem, see) that any can be expressed this way: i.e., that is onto. Since is onto and isometric, it is an isomorphism of Banach spaces. With this (isometric) isomorphism in mind, it is usual to say simply that is the continuous dual space of For the space is reflexive. Let be as above and let be the corresponding linear isometry. Consider the map from to obtained by composing with the transpose (or adjoint) of the inverse of This map coincides with the canonical embedding of into its bidual. Moreover, the map is onto, as composition of two onto isometries, and this proves reflexivity. If the measure on is sigma-finite, then the dual of is isometrically isomorphic to (more precisely, the map corresponding to is an isometry from onto The dual of is subtler. Elements of can be identified with bounded signed finitely additive measures on that are absolutely continuous with respect to See ba space for more details. If we assume the axiom