4.2. HILBERTSPACE 51
f(x),−∞<x<∞representthe”components”of|f>,andf∗(x)representsthe
componentsofthecorrespondingbra<f|.
Thereisalinearoperationcalledmatrixmultiplicationwhichturnsavectorinto
anothervector
%v′=M%v (4.24)
or
|v′>=M|v> (4.25)
inournewnotation.Intermsofcomponents,matrixmultiplicationisdefinedas
v′i=
∑N
j=1
Mijvj (4.26)
andithasthelinearitypropertythat
M(a|u>+b|v>)=aM|u>+bM|v> (4.27)
whereaandbareconstants. Thereisasimilarlinearoperationwhichturnsfunctions
intootherfunctions
|f′>=O|f> (4.28)
havingthelinearityproperty
O(a|f>+b|g>)=aO|f>+bO|g> (4.29)
Intermsof”components,”thisoperationiswritten
f′(x)=
∫∞
−∞
dyO(x,y)f(y) (4.30)
whereO(x,y)issomefunctionoftwovariablesxandy,incompleteanalogytothe
ruleformatrixmultiplication(4.26). Finally,theexpressioninlinearalgebra
<u|M|v> = %u·M%v
=
∑N
i=1
∑N
j=1
u∗iMijvj (4.31)
corresponds,inthecaseoffunctions,totheexpression
<g|O|f>=
∫∞
−∞
dx
∫∞
−∞
dyg∗(x)O(x,y)f(y) (4.32)
Afunction,therefore, isjustavectorwithacontinuousindex. Sincethereare
aninfinitenumberof”components”(onecomponentf(x)foreachvalueofx),the
vectorisamemberofaninfinite-dimensionalspaceknownas”HilbertSpace.”Stated
anotherway: Hilbert Space is theinfinite-dimensional vectorspace of all
square-integrablefunctions.