nLab
inner product space

Idea

An inner product space is a vector space $V$ equipped with a (conjugate)-symmetric bilinear or sesquilinear form: a linear map from $V\otimes V$ or $\overline{V}\otimes V$ to the ground ring $k$.

One often studies positive-definite inner product spaces; for these, see Hilbert space. Here we do not assume positivity or definiteness. On the other hand, it is very common to assume, although perhaps not universally assumed, that the form is nondegenerate; see also bilinear form.

Definitions

Let $V$ be a vector space over the field (or more generally a ring) $k$. Suppose that $k$ is equipped with an involution $r\mapsto \overline{r}$, called conjugation; in many examples, this will simply be the identity function, but not always. An inner product on $V$ is a function

$$⟨-,-⟩:V\times V\to k$$\langle {-},{-} \rangle: V \times V \to k

that is (1–3) sesquilinear (or bilinear when the involution is the identity) and (4) conjugate-symmetric (or symmetric when the involution is the identity). That is:

1. $⟨0,x⟩=0$ and $⟨x,0⟩=0$;
2. $⟨x+y,z⟩=⟨x,z⟩+⟨y,z⟩$ and $⟨x,y+z⟩=⟨x,y⟩+⟨x,z⟩$;
3. $⟨cx,y⟩=\overline{c}⟨x,y⟩$ and $⟨x,cy⟩=⟨x,y⟩c$;
4. $⟨x,y⟩=\overline{⟨y,x⟩}$.

Here we use the physicist's convention that the inner product is antilinear ( = conjugate-linear) in the first variable rather than in the second, rather than the mathematician's convention, which is the reverse. The physicist's convention fits in a little better with $2$-Hilbert spaces and is often used in a generalization for Hilbert modules. Note that we use the same ring as values of the inner product as for scalars. Notice that $⟨x,cy⟩=⟨x,y⟩c$ is written with $c$ on the right for the case that we deal with noncommutative division ring.

Are the two conventions really equivalent when $k$ is noncommutative? —Toby

(The axiom list above is rather redundant. First of all, (1) follows from (3) by setting $c=0$; besides that, (1–3) come in pairs, only one of which is needed, since each half follows from the other using (4). It is even possible to derive (3) from (2) under some circumstances.)

An inner product space is simply a vector space equipped with an inner product.

Conditions on inner products

We define a function $\parallel -{\parallel }^2:V\to k$ by $\parallel x{\parallel }^2=⟨x,x⟩$; this is called the norm of $x$. As the notation suggests, it is common to take the norm of $x$ to be the square root of this expression in contexts where that makes sense, but for us $\parallel -{\parallel }^2$ is an atomic symbol. Note that the norm of $x$ is real in that it equals its own conjugate, by (4). Then we can consider some conditions on the inner product:

• Notice that (by 1), $\parallel x{\parallel }^2=0$ if $x=0$; the inner product is definite if the converse holds.
• More generally, the inner product is semidefinite if …
• On the other hand, the inner product is indefinite if …

Semidefinite inner products behave very much like definite ones; you can mod out by the elements with norm $0$ to get a quotient space with a definite inner product.

Now suppose that $k$ (or at least, its subring of real values) is an ordered field. Then we can consider other conditions on the inner product:

• The inner product is positive semidefinite, or simply positive, if $\parallel x{\parallel }^2\ge 0$ always.
• The inner product is positive definite if it is both positive and definite, in other words if $\mid x{\mid }^2>0$ whenever $x\ne 0$.
• The inner product is negative semidefinite, or simply negative, if $\parallel x{\parallel }^2\le 0$ always.
• The inner product is negative definite if it is both positive and definite, in other words if $\mid x{\mid }^2<0$ whenever $x\ne 0$.

In this case, we have these theorems:

• A semidefinite inner product is either positive or negative.
• A definite inner product is either positive or negative definite.
• An inner product is indefinite if and only if some norms are positive and some are negative.

Negative (semi)definite inner products behave very much like positive (semi)definite ones; you can turn one into the other by multiplying all inner products by $-1$.

The study of positive definite inner product spaces (hence essentially of all semidefinite inner product spaces over ordered fields) is essentially the study of Hilbert spaces. (For Hilbert spaces, one usually uses a topological field, typically $ℂ$, and requires a completeness condition, but this does not effect the algebraic properties much.) The study of indefinite inner product spaces is very different; see the English Wikipedia article on Krein space?s for some of it.