Page 113 - Linear Algebra Done Right

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v ∈ V;
v, v ð 0 for all
If z is a complex
number, then the positivity Chapter 6. Inner-Product Spaces
statement z ≥ 0 means definiteness
v, v = 0 if and only if v = 0;
that z is real and
nonnegative.
additivity in first slot
u + v, w = u, w + v, w for all u, v, w ∈ V;
homogeneity in first slot
av, w = a v, w for all a ∈ F and all v, w ∈ V;
conjugate symmetry
v, w = w, v for all v, w ∈ V.

Recall that every real number equals its complex conjugate. Thus
if we are dealing with a real vector space, then in the last condition
above we can dispense with the complex conjugate and simply state
that v, w = w, v for all v, w ∈ V.
An inner-product space is a vector space V along with an inner
product on V.
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The most important example of an inner-product space is F .We
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If we are dealing with can define an inner product on F by
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R rather than C , then
again the complex 6.1 (w 1 ,...,w n ), (z 1 ,...,z n ) = w 1 z 1 + ··· + w n z n ,
conjugate can be
as you should verify. This inner product, which provided our motiva-
ignored.
tion for the definition of an inner product, is called the Euclidean inner
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product on F . When F is referred to as an inner-product space, you
should assume that the inner product is the Euclidean inner product
unless explicitly told otherwise.
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There are other inner products on F in addition to the Euclidean
inner product. For example, if c 1 ,...,c n are positive numbers, then we
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can define an inner product on F by
(w 1 ,...,w n ), (z 1 ,...,z n ) = c 1 w 1 z 1 +· · ·+ c n w n z n ,
as you should verify. Of course, if all the c’s equal 1, then we get the
Euclidean inner product.
As another example of an inner-product space, consider the vector
space P m (F) of all polynomials with coefficients in F and degree at
most m. We can define an inner product on P m (F) by