Page 227 - A Course in Linear Algebra with Applications

7.2: Inner Product Spaces 211

since f(x) 2 > 0; also, if we think of the integral as the area
2
under the curve y = f{x) , then it becomes clear that the
integral cannot vanish unless f(x) is identically equal to zero
in [a ,b).

Example 7.2.3
Define an inner product on the vector space P n (R) of all real
polynomials in x of degree less than n by the rule

n
< f,9> = ^2f(.Xi)g(xi)
i=l
where distinct real numbers.
Here it is not so clear why the first requirement for an
inner product holds. Note that

n

also the only way that this sum can vanish is if f(x\) = ...
x
= f( n) = 0. But / is a polynomial of degree at most n — 1, so
it cannot have n distinct roots unless it is the zero polynomial.

Orthogonality in inner product spaces

A real inner product space is a vector space V over R
together with an inner product < > on V. It will be con-
n
venient to speak of "the inner product space V , suppressing
mention of the inner product where this is understood. Thus
n
"the inner product space R " refers to R n with the scalar
product as inner product: this is called the Euclidean inner
product space.
Two vectors u and v of an inner product space V are said
to be orthogonal if
< u, v > = 0.