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# Inner product space

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Let V be a vector space.

An inner product on V is a map f(x,y): V x V → F where F is the root field (either R or C) subject to the following axioms:

1. for any x in V, f(x,x)≥0 and f(x,x)=0 if and only if x=0
2. f(a,x+y,z)=af(x,z)+f(y,z) for any a in F, x,y in V.
3. f(x,y)=f*(y,x) where * denotes complex conjugation whenever x,y are in V.

Here and in the sequel, we will write <x,y> for f(x,y) and ||x|| for √<x,x>. The latter is well defined by axiom 1.

From these axioms, we can do nothing but conclude the following:

- Theorem (Cauchy-Schwarz): |<x,y>|≤||x||·||y|| for any x,y in V
- Theorem (Triangle Inequality): ||x+y||≤||x||+||y||
- Theorem (Pythagoras): ||x||2+||y||2=||x+y||2 whenever x,y are in V and <x,y>=0.

An induction on Pythagoras yields:

- Theorem (Pythagoras): ∑ ||xk||2 = ||∑xk||2 whenever x1, ..., xn are orthogonal vectors, that is, <xj,xk>=0 whenever j≠k.

Because of the triangle inequality and because of axiom 2, we see that ||·|| is an honest to goodness norm (see normed vector space).

In view of the Cauchy-Schwarz inequality, we also note that <·,·> is continuous from V x V to F. This allows us to extend Pythagoras' theorem a tiny bit more, and rename it:

Theorem (Parseval's Identity): If xk are vectors in V and if ∑vk converges to x in V, and if the vectors xk are mutually orthogonal, then

- ∑||xk||2=||x||2

A homomorphism of inner product spaces is a simple linear map. If it were to preserve the inner product, it would become an isometry -- though in a strict sense, there is no reason to dislike that idea. An isomorphism of inner product spaces is normally a linear map that preserves the inner product. Such an isomorphism is also called an isometry. In certain fields, "isomorphism" refers strictly to the vector space structure, in which case an inner product preserving linear map will be termed "isometrically isomorphic". Note that it is sufficient for a linear map to preserve the norm, for it to preserve the inner product. Indeed, via the polarization identities <u+v,u+v>=<u,u>+2R<u,v>+<v,v> (where R is the real part) and <iu+v,iu+v>=<v,v>-<u,u>-2I<u,v> (where I is the imaginary part), we can recover the inner product from the norm.

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Last edited August 15, 2001 10:31 am (diff)
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