- #1

SrEstroncio

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In an effort to keep me from spending all summer lying on the couch, I recently started reading Michael Spivak's Calculus on Manifolds; while working on problem 1-6 I got stuck on a technical detail and I was wondering if anyone could provide a little insight.

Problem 1-6 says:

Let [tex] f [/tex] and [tex] g [/tex] be integrable functions on [tex] [a,b] [/tex].

Prove that [tex] |\int_a^b fg | \leq (\int_a^b f^2)^{1/2}(\int_a^b g^2)^{1/2} [/tex].

He suggests that you treat the cases [tex] 0=\int_a^b (f-\lambda g)^2 [/tex] for some [tex] \lambda \in R [/tex] and [tex] 0 \lt \int_a^b (f-\lambda g)^2 [/tex] for all [tex] \lambda [/tex] separately.

My question is: how do I know the [tex] \lambda [/tex] is unique?

Considering the two cases given above I got a cuadratic expression in [tex] \lambda [/tex] whose discriminant gave me the strict inequality when [tex] 0 \lt \int_a^b (f-\lambda g)^2 [/tex] for all [tex] \lambda [/tex] (since there are no real roots of the equation), but in order to conclude that [\tex] |\int_a^b fg | \leq (\int_a^b f^2)^{1/2}(\int_a^b g^2)^{1/2} [/tex] I am forced to assume that the discriminant of the equation is equal to zero (otherwise I get [tex] |\int_a^b fg | \geq (\int_a^b f^2)^{1/2}(\int_a^b g^2)^{1/2} [/tex], which is obviously wrong), meaning that there is only one root of the equation, or equivalently that the lambda that satisfies [tex] 0=\int_a^b (f-\lambda g)^2 [/tex] is unique, fact that I feel must be proven, not assumed).

How do I know said lambda is unique? Keep in mind that since f and g are integrable (but may not be continuous) one cannot assume that [tex] 0=\int_a^b (f)^2 [/tex] implies [tex] f=0 [/tex].

Problem 1-6 says:

Let [tex] f [/tex] and [tex] g [/tex] be integrable functions on [tex] [a,b] [/tex].

Prove that [tex] |\int_a^b fg | \leq (\int_a^b f^2)^{1/2}(\int_a^b g^2)^{1/2} [/tex].

He suggests that you treat the cases [tex] 0=\int_a^b (f-\lambda g)^2 [/tex] for some [tex] \lambda \in R [/tex] and [tex] 0 \lt \int_a^b (f-\lambda g)^2 [/tex] for all [tex] \lambda [/tex] separately.

My question is: how do I know the [tex] \lambda [/tex] is unique?

Considering the two cases given above I got a cuadratic expression in [tex] \lambda [/tex] whose discriminant gave me the strict inequality when [tex] 0 \lt \int_a^b (f-\lambda g)^2 [/tex] for all [tex] \lambda [/tex] (since there are no real roots of the equation), but in order to conclude that [\tex] |\int_a^b fg | \leq (\int_a^b f^2)^{1/2}(\int_a^b g^2)^{1/2} [/tex] I am forced to assume that the discriminant of the equation is equal to zero (otherwise I get [tex] |\int_a^b fg | \geq (\int_a^b f^2)^{1/2}(\int_a^b g^2)^{1/2} [/tex], which is obviously wrong), meaning that there is only one root of the equation, or equivalently that the lambda that satisfies [tex] 0=\int_a^b (f-\lambda g)^2 [/tex] is unique, fact that I feel must be proven, not assumed).

How do I know said lambda is unique? Keep in mind that since f and g are integrable (but may not be continuous) one cannot assume that [tex] 0=\int_a^b (f)^2 [/tex] implies [tex] f=0 [/tex].

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