Page 452 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition
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Calculus Required
Remark Since polynomials are continuous functions on , they are continuous on any closed interval . Thus,
for all such intervals the vector space is a subspace of , and Formula 6 defines an inner product on .
Calculus Required
Remark Recall from calculus that the arc length of a curve over an interval is given by the formula
Do not confuse this concept of arc length with , which is the length (norm) of f when f is viewed as a vector in (8)
Formulas 7 and 8 are quite different. .
The following theorem lists some basic algebraic properties of inner products.
THEOREM 6.1.1
Properties of Inner Products
If u, v, and w are vectors in a real inner product space, and k is any scalar, then
(a)
(b)
(c)
(d)
(e)
Proof We shall prove part and leave the proofs of the remaining parts as exercises.
