Editing Inner product space (section)

== Definition ==
In this article, {{math|''F''}} denotes a [[field (mathematics)|field]] that is either the [[real number]]s <math>\R,</math> or the [[complex number]]s <math>\Complex.</math> A [[scalar (mathematics)|scalar]] is thus an element of {{math|''F''}}. A bar over an expression representing a scalar denotes the [[complex conjugate]] of this scalar. A zero vector is denoted <math>\mathbf 0</math> for distinguishing it from the scalar {{math|0}}.

An ''inner product'' space is a [[vector space]] {{math|''V''}} over the field {{math|''F''}} together with an ''inner product'', that is, a map
<math display="block"> \langle \cdot, \cdot \rangle : V \times V \to F </math>
that satisfies the following three properties for all vectors <math>x,y,z\in V</math> and all scalars {{nowrap|<math>a,b \in F</math>.<ref name= Jain>{{cite book |title=Functional Analysis |first1=P. K. |last1=Jain |first2=Khalil |last2=Ahmad |chapter-url=https://books.google.com/books?id=yZ68h97pnAkC&pg=PA203 |page=203 |chapter=5.1 Definitions and basic properties of inner product spaces and Hilbert spaces |isbn=81-224-0801-X |year=1995 |edition=2nd |publisher=New Age International}}</ref><ref name="Prugovec̆ki">{{cite book |title=Quantum Mechanics in Hilbert Space |first=Eduard |last=Prugovečki |chapter-url=https://books.google.com/books?id=GxmQxn2PF3IC&pg=PA18 |chapter=Definition 2.1 |pages=18ff |isbn=0-12-566060-X | year = 1981 |publisher=Academic Press |edition = 2nd}}</ref>}}

* ''Conjugate symmetry'': <math display=block>\langle x, y \rangle = \overline{\langle y, x \rangle}.</math> As <math display="inline"> a = \overline{a} </math> [[if and only if]] <math>a</math> is real, conjugate symmetry implies that <math>\langle x, x \rangle </math> is always a real number. If {{math|''F''}} is <math>\R</math>, conjugate symmetry is just symmetry.
* [[Linear map|Linearity]] in the first argument:<ref group="Note">By combining the ''linear in the first argument'' property with the ''conjugate symmetry'' property you get ''conjugate-linear in the second argument'': <math display="inline"> \langle x,by \rangle = \langle x,y \rangle \overline{b} </math>.  This is how the inner product was originally defined and is used in most mathematical contexts. A different convention has been adopted in theoretical physics and quantum mechanics, originating in the [[bra-ket]] notation of [[Paul Dirac]], where the inner product is taken to be ''linear in the second argument'' and ''conjugate-linear in the first argument''; this convention is used in many other domains such as engineering and computer science.</ref> <math display=block>
     \langle ax+by, z \rangle = a \langle x, z \rangle + b  \langle y, z \rangle.</math>
* [[Definite bilinear form|Positive-definiteness]]: if <math>x</math> is not zero, then <math display=block>
      \langle x, x \rangle > 0
</math> (conjugate symmetry implies that <math>\langle x, x \rangle</math> is real).

If the positive-definiteness condition is replaced by merely requiring that <math>\langle x, x \rangle \geq 0</math> for all <math>x</math>, then one obtains the definition of ''positive semi-definite Hermitian form''. A positive semi-definite Hermitian form <math>\langle \cdot, \cdot \rangle</math> is an inner product if and only if for all <math>x</math>, if <math>\langle x, x \rangle = 0</math> then <math>x = \mathbf 0</math>.{{sfn|Schaefer|Wolff|1999|p=44}}

=== Basic properties ===
In the following properties, which result almost immediately from the definition of an inner product, {{math|''x'', ''y''}} and {{mvar|z}} are arbitrary vectors, and {{mvar|a}} and {{mvar|b}} are arbitrary scalars. 
*<math>\langle \mathbf{0}, x \rangle=\langle x,\mathbf{0}\rangle=0.</math>
*<math> \langle x, x \rangle</math> is real and nonnegative.
*<math>\langle x, x \rangle = 0</math> if and only if <math>x=\mathbf{0}.</math>
*<math>\langle x, ay+bz \rangle= \overline a \langle x, y \rangle + \overline b \langle x, z \rangle.</math><br>This implies that an inner product is a [[sesquilinear form]].
*<math>\langle x + y, x + y \rangle = \langle x, x \rangle + 2\operatorname{Re}(\langle x, y \rangle) + \langle y, y \rangle,</math> where <math>\operatorname{Re}</math><br>denotes the [[real part]] of its argument.

Over <math>\R</math>, conjugate-symmetry reduces to symmetry, and sesquilinearity reduces to bilinearity. Hence an inner product on a real vector space is a ''positive-definite symmetric [[bilinear form]]''. The [[binomial expansion]] of a square becomes
<math display="block">\langle x + y, x + y \rangle = \langle x, x \rangle + 2\langle x, y \rangle + \langle y, y \rangle .</math>

=== Notation ===

Several notations are used for inner products, including 
<math> \langle \cdot, \cdot \rangle </math>, 
<math> \left ( \cdot, \cdot \right ) </math>,
<math> \langle \cdot | \cdot \rangle </math> and
<math> \left ( \cdot | \cdot \right ) </math>, as well as the usual dot product.

=== Convention variant ===

Some authors, especially in [[physics]] and [[matrix algebra]], prefer to define inner products and sesquilinear forms with linearity in the second argument rather than the first. Then the first argument becomes conjugate linear, rather than the second. [[Bra–ket notation|Bra-ket notation]] in [[quantum mechanics]] also uses slightly different notation, i.e. <math> \langle \cdot | \cdot \rangle </math>, where <math> \langle x | y \rangle := \left ( y, x \right ) </math>.