Editing Riesz representation theorem (section)

=== Relationship with the associated real Hilbert space ===
{{See also|Complexification}}

Assume that <math>H</math> is a complex Hilbert space with inner product <math>\langle \,\cdot\mid\cdot\, \rangle.</math> 
When the Hilbert space <math>H</math> is reinterpreted as a real Hilbert space then it will be denoted by <math>H_{\R},</math> where the (real) inner-product on <math>H_{\R}</math> is the real part of <math>H</math>'s inner product; that is: 
<math display=block>\langle x, y \rangle_{\R} := \operatorname{re} \langle x, y \rangle.</math> 

The norm on <math>H_{\R}</math> induced by <math>\langle \,\cdot\,, \,\cdot\, \rangle_{\R}</math> is equal to the original norm on <math>H</math> and the continuous dual space of <math>H_{\R}</math> is the set of all {{em|real}}-valued bounded <math>\R</math>-linear functionals on <math>H_{\R}</math> (see the article about the [[polarization identity]] for additional details about this relationship). 
Let <math>\psi_{\R} := \operatorname{re} \psi</math> and <math>\psi_{i} := \operatorname{im} \psi</math> denote the real and imaginary parts of a linear functional <math>\psi,</math> so that <math>\psi = \operatorname{re} \psi + i \operatorname{im} \psi = \psi_{\R} + i \psi_{i}.</math> 
The formula [[Real and imaginary parts of a linear functional|expressing a linear functional]] in terms of its real part is
<math display=block>\psi(h) = \psi_{\R}(h) - i \psi_{\R} (i h) \quad \text{ for } h \in H,</math>
where <math>\psi_{i}(h) = - i \psi_{\R} (i h)</math> for all <math>h \in H.</math> 
It follows that <math>\ker\psi_{\R} = \psi^{-1}(i \R),</math> and that <math>\psi = 0</math> if and only if <math>\psi_{\R} = 0.</math> 
It can also be shown that <math>\|\psi\| = \left\|\psi_{\R}\right\| = \left\|\psi_i\right\|</math> where <math>\left\|\psi_{\R}\right\| := \sup_{\|h\| \leq 1} \left|\psi_{\R}(h)\right|</math> and <math>\left\|\psi_i\right\| := \sup_{\|h\| \leq 1} \left|\psi_i(h)\right|</math> are the usual [[operator norm]]s. 
In particular, a linear functional <math>\psi</math> is bounded if and only if its real part <math>\psi_{\R}</math> is bounded. 

'''Representing a functional and its real part'''

The Riesz representation of a continuous linear function <math>\varphi</math> on a complex Hilbert space is equal to the Riesz representation of its real part <math>\operatorname{re} \varphi</math> on its associated real Hilbert space. 

Explicitly, let <math>\varphi \in H^*</math> and as above, let <math>f_\varphi \in H</math> be the Riesz representation of <math>\varphi</math> obtained in <math>(H, \langle, \cdot, \cdot \rangle),</math> so it is the unique vector that satisfies <math>\varphi(x) = \left\langle f_{\varphi} \mid x \right\rangle</math> for all <math>x \in H.</math> 
The real part of <math>\varphi</math> is a continuous real linear functional on <math>H_{\R}</math> and so the Riesz representation theorem may be applied to <math>\varphi_{\R} := \operatorname{re} \varphi</math> and the associated real Hilbert space <math>\left(H_{\R}, \langle, \cdot, \cdot \rangle_{\R}\right)</math> to produce its Riesz representation, which will be denoted by <math>f_{\varphi_{\R}}.</math> 
That is, <math>f_{\varphi_{\R}}</math> is the unique vector in <math>H_{\R}</math> that satisfies <math>\varphi_{\R}(x) = \left\langle f_{\varphi_{\R}} \mid x \right\rangle_{\R}</math> for all <math>x \in H.</math> 
The conclusion is <math>f_{\varphi_{\R}} = f_{\varphi}.</math> 
This follows from the main theorem because <math>\ker\varphi_{\R} = \varphi^{-1}(i \R)</math> and if <math>x \in H</math> then 
<math display=block>\left\langle f_\varphi \mid x \right\rangle_{\R} = \operatorname{re} \left\langle f_\varphi \mid x \right\rangle = \operatorname{re} \varphi(x) = \varphi_{\R}(x)</math> 
and consequently, if <math>m \in \ker\varphi_{\R}</math> then <math>\left\langle f_{\varphi}\mid m \right\rangle_{\R} = 0,</math> which shows that <math>f_{\varphi} \in (\ker\varphi_{\R})^{\perp_{\R}}.</math> 
Moreover, <math>\varphi(f_\varphi) = \|\varphi\|^2</math> being a real number implies that <math>\varphi_{\R} (f_\varphi) = \operatorname{re} \varphi(f_\varphi) = \|\varphi\|^2.</math>
In other words, in the theorem and constructions above, if <math>H</math> is replaced with its real Hilbert space counterpart <math>H_{\R}</math> and if <math>\varphi</math> is replaced with <math>\operatorname{re} \varphi</math> then <math>f_{\varphi} = f_{\operatorname{re} \varphi}.</math> This means that vector <math>f_{\varphi}</math> obtained by using <math>\left(H_{\R}, \langle, \cdot, \cdot \rangle_{\R}\right)</math> and the real linear functional <math>\operatorname{re} \varphi</math> is the equal to the vector obtained by using the origin complex Hilbert space <math>\left(H, \left\langle, \cdot, \cdot \right\rangle\right)</math> and original complex linear functional <math>\varphi</math> (with identical norm values as well). 

Furthermore, if <math>\varphi \neq 0</math> then <math>f_{\varphi}</math> is perpendicular to <math>\ker\varphi_{\R}</math> with respect to <math>\langle \cdot, \cdot \rangle_{\R}</math> where the kernel of <math>\varphi</math> is be a ''proper'' subspace of the kernel of its real part <math>\varphi_{\R}.</math> Assume now that <math>\varphi \neq 0.</math> 
Then <math>f_{\varphi} \not\in \ker\varphi_{\R}</math> because <math>\varphi_{\R}\left(f_{\varphi}\right) = \varphi\left(f_{\varphi}\right) = \|\varphi\|^2 \neq 0</math> and <math>\ker\varphi</math> is a proper subset of <math>\ker\varphi_{\R}.</math> The vector subspace <math>\ker \varphi</math> has real codimension <math>1</math> in <math>\ker\varphi_{\R},</math> while <math>\ker\varphi_{\R}</math> has {{em|real}} codimension <math>1</math> in <math>H_{\R},</math> and <math>\left\langle f_{\varphi}, \ker\varphi_{\R} \right\rangle_{\R} = 0.</math> That is, <math>f_{\varphi}</math> is perpendicular to <math>\ker\varphi_{\R}</math> with respect to <math>\langle \cdot, \cdot \rangle_{\R}.</math>