Number Theory: An Introduction to Mathematics

10 Inner Product Spaces 71

which impliesaib=bai(i= 1 ,...,n). Since this holds for allb∈MandMis a
maximal subfield ofD, it follows thatai∈M(i= 1 ,...,n).
LetT denote the set of all linear transformations ofDas a vector space overM.
By what we have already proved, everyT ∈T is anM-linear combination of the
mapsT 1 ,...,Tn,whereTix=xei(i= 1 ,...,n), and the mapsT 1 ,...,Tnare lin-
early independent overM. Consequently the dimension ofTas a vector space overM
isn.ButT has dimension [D:M]^2 as a vector space overM. Hence [D:M]^2 =n.
Sincen=[D:M][M:C], it follows that [D:M]=[M:C].

10 Inner Product Spaces........................................

LetFdenote either the real fieldRor the complex fieldC. A vector spaceVoverFis
said to be aninner product spaceif there exists a map(u,v)→〈u,v〉ofV×Vinto
Fsuch that for everyα∈Fand allu,u′,v∈V,

(i)〈αu,v〉=α〈u,v〉,
(ii)〈u+u′,v〉=〈u,v〉+〈u′,v〉,
(iii)〈v,u〉=〈u,v〉,
(iv)〈u,u〉>0ifu=O.

IfF=R, then (iii) simply says that〈v,u〉=〈u,v〉, since a real number is its own
complex conjugate. The restrictionu=Ois necessary in (iv), since (i) and (iii) imply
that

〈u,O〉=〈O,v〉=0forallu,v∈V.

It follows from (ii) and (iii) that

〈u,v+v′〉=〈u,v〉+〈u,v′〉 for allu,v,v′∈V,

and from (i) and (iii) that

〈u,αv〉= ̄α〈u,v〉 for everyα∈Fand allu,v∈V.

The standard example of an inner product space is the vector spaceFn, with the
inner product ofx=(ξ 1 ,...,ξn)andy=(η 1 ,...,ηn)defined by

〈x,y〉=ξ 1 η ̄ 1 +···+ξnη ̄n.

Another example is the vector spaceC(I)of all continuous functionsf :I →F,
whereI =[a,b] is a compact subinterval ofR, with the inner product of fandg
defined by

〈f,g〉=

∫b

a

f(t)g(t)dt.

In an arbitrary inner product spaceVwe define thenorm‖v‖of a vectorv∈Vby

‖v‖=〈v,v〉^1 /^2.