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# inner product

An *inner product* on a vector space $V$ over a field $K$ (which must be either the field $\mathbb{R}$ of real numbers or the field $\mathbb{C}$ of complex numbers) is a function $(\ ,\ ):V\times V\longrightarrow K$ such that, for all $k_{1},k_{2}\in K$ and ${{\bf v}}_{1},{{\bf v}}_{2},{{\bf v}},{{\bf w}}\in V$, the following properties hold:

1. $(k_{1}{{\bf v}}_{1}+k_{2}{{\bf v}}_{2},{{\bf w}})=k_{1}({{\bf v}}_{1},{{\bf w}% })+k_{2}({{\bf v}}_{2},{{\bf w}})$ (linearity

^{1}^{1}A small minority of authors impose linearity on the second coordinate instead of the first coordinate.)2. $({{\bf v}},{{\bf w}})=\overline{({{\bf w}},{{\bf v}})}$, where $\overline{\ \ \ \ }$ denotes complex conjugation (conjugate symmetry)

3. $({{\bf v}},{{\bf v}})\geq 0$, and $({{\bf v}},{{\bf v}})=0$ if and only if ${{\bf v}}={{\bf 0}}$ (positive definite)

(Note: Rule 2 guarantees that $({{\bf v}},{{\bf v}})\in\mathbb{R}$, so the inequality $({{\bf v}},{{\bf v}})\geq 0$ in rule 3 makes sense even when $K=\mathbb{C}$.)

The standard example of an inner product is the dot product on $K^{n}$:

$((x_{1},\dots,x_{n}),(y_{1},\dots,y_{n})):=\sum_{{i=1}}^{n}x_{i}\overline{y_{i}}$ |

Every inner product space is a normed vector space, with the norm being defined by $||{{\bf v}}||:=\sqrt{({{\bf v}},{{\bf v}})}$.

## Mathematics Subject Classification

11E39*no label found*15A63

*no label found*

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## Comments

## parens

Note that half of the time, <.,.> is used to denote an inner product, instead of parens.

-apk

## Re: parens

You are correct.By the way the symbols <> are called "angle brackets"

and () parentesis

## Re: parens

1. Physicists generally use angle brackets with a vertical bar instead of the comma: <.|.>. This is the inner product in Dirac's bra-ket notation.

2. From the footnote: "A small minority of authors impose linearity on the second coordinate instead of the first coordinate." Perhaps, if "authors" is replaced with "mathematicians". Essentially all, if not all, physics texts impose linearity on the second coordinate.

## Re: parens

> 1. Physicists generally use angle brackets with a vertical

> bar instead of the comma: <.|.>. This is the inner product

> in Dirac's bra-ket notation.

Yes, although that is because they use |.> and <.| in separation,

the former usually for elements of a vector space and the latter

for functionals on that vector space, so they "really" leave one

of the bars out when they apply functional to vector.

> 2. From the footnote: "A small minority of authors impose

> linearity on the second coordinate instead of the first

> coordinate." Perhaps, if "authors" is replaced with

> "mathematicians". Essentially all, if not all, physics

> texts impose linearity on the second coordinate.

Since this footnote appears in the entry for something as basic

as "inner product", the linearity business really has to be

explained more fully. It's only an issue for complex vector

spaces, so maybe the definition is best stated first for real

vector spaces (noting that the inner product is linear in both

factors), and then again for complex vector spaces (with a remark

that it has to be linear in one factor and conjugate linear (?)

in the other, noting various conventions on the matter).

What's more, I have seen definitions of inner products also

over fields of positive characteristic, so the claim that inner

products exist only over the reals and complex numbers is

disputable.

## Re: parens

> the claim that inner products exist only over the reals and complex

> numbers is disputable.

In order to formulate the notion of an inner product, the only structure one need is the notion of field and the notion of inner product --- put another way, the only terms needed to frame the definition are the primitive terms which occur in the definitions of fields and of vector spaces. Therefore, a restriction that states inner products only exist over the real and complex fields would be artificial. Since nothing in the definition of an inner product requires us to impose restrictions on the field, Ockham's razor says that we should not do so.

In order to formulate the notion of a conjugate-linear inner product, all one requires is an automorphism of order 2. The best known example of such an automorphism is complex conjugation, which is why one usually deals with conjugate-linear inner products over the complex number field, but there is nothing in the definition which inherently limits one to the complex numbers.

Therefore, I would advise that you formulate your definitions in a fashion which does not impose any restriction on the base field except for the requrement that there exist a conjugation of order 2 in the case of the definition of a conjugate linear inner product. Later on, you could mention that the most common case is that of the vector spaces over the fields of real and complex numbers.