identity for additional details about this relationship). Let and denote the real and imaginary parts of a linear functional so that The formula expressing a linear functional in terms of its real part is where for all It follows that and that if and only if It can also be shown that where and are the usual operator norms. In particular, a linear functional is bounded if and only if its real part is bounded. Representing a functional and its real part The Riesz representation of a continuous linear function on a complex Hilbert space is equal to the Riesz representation of its real part on its associated real Hilbert space. Explicitly, let and as above, let be the Riesz representation of obtained in so it is the unique vector that satisfies for all The real part of is a continuous real linear functional on and so the Riesz representation theorem may be applied to and the associated real Hilbert space to produce its Riesz representation, which will be denoted by That is, is the unique vector in that satisfies for all The conclusion is This follows from the main theorem because and if then and consequently, if then which shows that Moreover, being a real number implies that In other words, in the theorem and constructions above, if is replaced with its real Hilbert space counterpart and if is replaced with then This means that vector obtained by using and the real linear functional is the equal to the vector obtained by using the origin complex Hilbert space and original complex linear functional (with identical norm values as well). Furthermore, if then is perpendicular to with respect to where the kernel of is be a proper subspace of the kernel of its real part Assume now that Then because and is a proper subset of The vector subspace has real codimension in while has codimension in and That is, is perpendicular to with respect to Canonical injections into the dual and anti-dual Induced linear map into anti-dual The map defined by placing into the coordinate of the inner product and letting the variable vary over the coordinate results in an functional: This map is an element of which is the continuous anti-dual space of The is the operator which is also an injective isometry. The Fundamental theorem of Hilbert spaces, which is related to Riesz representation theorem, states that