Question

The following are all manifestations of the Cauchy - Schwarz Inequality. In other words, we can show they are true becaue we can find a vector space V and an inner product on V, such that the conclusion of the Cauchy - Schwarz Inequality is the indicated inequalit. Your job is to prove these inequalities by finding the vector space V and the inner product on V, and then unwinding the conclusion of the Cauchy - Schwarz Inequality to obtain the indicated inequality. Do not try to prove these inequalities directly.

[integral(f(x) g(x)) from a to b dx]^2 <=[integral([f(x)]^2) from a to b dx][i

Answer 1

The definition of a vector space has Cauchy Schwarz behaviour as a resultant. What is being effectively asked is whether some given integration formula can satisfy the required semantics for inner product. One has to demonstrate those semantics.

There are many mathematical entities that yield inner product character. In this case integration represents a component wise multiplication and then accumulation as a sum. However it doesn't look like an inner product ( as generally first taught ) because the index set ( I assume here ) is real numbers generally and not integers specifically. Thus for geometric vectors X and Y in a Euclidean space of dimension N :
\[<X,Y> = \sum_{i=1}^{i=N} {x_{i}} {y_{i}}\]morphs to a space of integrable functions containing, say, two functions f and g\[<f,g> =\int\limits_{a}^{b}f(x)g(x)dx=\int\limits_{a}^{b}g(x)f(x)dx= <g,f>\]here I have used <> to denote inner product. To nail Cauchy Schwarz you'll need the norm of a vector/function to be non-negative, and this is true as :
\[<f,f> =\int\limits_{a}^{b}f(x)f(x)dx=\int\limits_{a}^{b}[f(x)]^{2}dx \]has a non-negative integrand for real valued functions. In spaces of complex valued functions the inner product is defined using the product of the complex conjugate of one function with the other. That is:
\[<f,g> =\int\limits_{a}^{b}f^{*}(x)g(x)dx \]and that ensures that \[f^{*}(x)f(x)\]is real and positive, and so is the summation/integral.

Thus it is the properties of Riemann integration that gives inner product semantics to these formulae. That in turn relies upon ( or requires ) said functions in the space - the f's and g's - to also have certain properties, specifically those that allow Riemann integration :
- must be defined and single valued on the interval [a,b]
- continuous on the closed interval [a,b]
if the integral is improper ( one or both limits is infinite ) then it must converge to a finite number.

NOW having said all of that, the inequality quoted in the question is not Cauchy Schwarz anyway. For the space of functions I have defined that would be :
\[[\int\limits_{a}^{b}f(x)g(x)dx]^{2}\le\int\limits_{a}^{b}[f(x)]^{2}dx\int\limits_{a}^{b}[g(x)]^{2}dx\]but I think the Equation scripting got truncated.