Editing Riesz representation theorem (section)

=== Mathematics vs. physics notations and definitions of inner product ===

The [[Hilbert space]] <math>H</math> has an associated [[inner product]] <math>H \times H \to \mathbb{F}</math> valued in <math>H</math>'s underlying scalar field <math>\mathbb{F}</math> that is linear in one coordinate and antilinear in the other (as specified below).
If <math>H</math> is a complex Hilbert space (<math>\mathbb{F} = \Complex</math>), then there is a crucial difference between the notations prevailing in mathematics versus physics, regarding which of the two variables is linear.
However, for real Hilbert spaces (<math>\mathbb{F} = \R</math>), the inner product is a [[Symmetric map|symmetric]] map that is linear in each coordinate ([[bilinear map|bilinear]]), so there can be no such confusion. 

In [[mathematics]], the inner product on a Hilbert space <math>H</math> is often denoted by <math>\left\langle \cdot\,, \cdot \right\rangle</math> or <math>\left\langle \cdot\,, \cdot \right\rangle_H</math> while in [[physics]], the [[bra–ket notation]] <math>\left\langle \cdot \mid \cdot \right\rangle</math> or <math>\left\langle \cdot \mid \cdot \right\rangle_H</math> is typically used. In this article, these two notations will be related by the equality:
<math display="block">\left\langle x, y \right\rangle := \left\langle y \mid x \right\rangle \quad \text{ for all } x, y \in H.</math>These have the following properties:<ol>
<li>The map <math>\left\langle \cdot\,, \cdot \right\rangle</math> is ''linear in its first coordinate''; equivalently, the map <math>\left\langle \cdot \mid \cdot \right\rangle</math> is ''linear in its second coordinate''. That is, for fixed <math>y \in H,</math> the map 
<math>\left\langle \,y\mid \cdot\, \right\rangle = \left\langle \,\cdot\,, y\, \right\rangle : H \to \mathbb{F}</math> with <math display="inline">h \mapsto \left\langle \,y\mid h\, \right\rangle = \left\langle \,h, y\, \right\rangle </math>
is a linear functional on <math>H.</math> This linear functional is continuous, so <math>\left\langle \,y\mid\cdot\, \right\rangle = \left\langle \,\cdot, y\, \right\rangle \in H^*.</math>
</li>
<li>The map <math>\left\langle \cdot\,, \cdot \right\rangle</math> is ''[[Antilinear map|antilinear]] in its {{em|second}} coordinate''; equivalently, the map <math>\left\langle \cdot \mid \cdot \right\rangle</math> is ''antilinear in its {{em|first}} coordinate''. That is, for fixed <math>y \in H,</math> the map
<math>\left\langle \,\cdot\mid y\, \right\rangle = \left\langle \,y, \cdot\, \right\rangle : H \to \mathbb{F}</math> with <math display="inline">h \mapsto \left\langle \,h\mid y\, \right\rangle = \left\langle \,y, h\, \right\rangle </math>
is an antilinear functional on <math>H.</math> This antilinear functional is continuous, so <math>\left\langle \,\cdot\mid y\, \right\rangle = \left\langle \,y, \cdot\, \right\rangle \in \overline{H}^*.</math>
</li>
</ol>

In computations, one must consistently use either the mathematics notation <math>\left\langle \cdot\,, \cdot \right\rangle</math>, which is (linear, antilinear); or the physics notation <math>\left\langle \cdot \mid \cdot \right\rangle</math>, which is (antilinear | linear).