is unfortunately the , meaning that the inner product is coordinate and antilinear in the other coordinate. This article will not choose one definition over the other. Instead, the assumptions made above make it so that the mathematics notation satisfies the mathematical convention/definition for the inner product (that is, linear in the first coordinate and antilinear in the other), while the physics bra–ket notation satisfies the physics convention/definition for the inner product (that is, linear in the second coordinate and antilinear in the other). Consequently, the above two assumptions makes the notation used in each field consistent with that field's convention/definition for which coordinate is linear and which is antilinear. Canonical norm and inner product on the dual space and anti-dual space If then is a non-negative real number and the map defines a canonical norm on that makes into a normed space. As with all normed spaces, the (continuous) dual space carries a canonical norm, called the , that is defined by The canonical norm on the (continuous) anti-dual space denoted by is defined by using this same equation: This canonical norm on satisfies the parallelogram law, which means that the polarization identity can be used to define a which this article will denote by the notations where this inner product turns into a Hilbert space. There are now two ways of defining a norm on the norm induced by this inner product (that is, the norm defined by ) and the usual dual norm (defined as the supremum over the closed unit ball). These norms are the same; explicitly, this means that the following holds for every As will be described later, the Riesz representation theorem can be used to give an equivalent definition of the canonical norm and the canonical inner product on The same equations that were used above can also be used to define a norm and inner product on 's anti-dual space Canonical isometry between the dual and antidual The complex conjugate of a functional which was defined above, satisfies for every and every This says exactly that the canonical antilinear bijection defined by as well as its inverse are antilinear isometries and consequently also homeomorphisms. The inner products on the dual space and the anti-dual space denoted respectively by and are related by and If then and this canonical map reduces down to the identity map. Riesz representation theorem Two vectors and are