Problem 6 - Calculus On Manifolds: A Modern Approach To Classical Theorems Of Advanced Calculus

Question

Let \(f\) and \(g\) be integrable on \([a, b]\) (a) Prove that \(\left|\int_{a}^{b} f \cdot g\right| \leq\left(\int_{a}^{b} f^{2}\right)^{\frac{1}{2}} \cdot\left(\int_{a}^{b} g^{2}\right)^{\frac{1}{2}} .\) Hint: Consider separately the cases \(0=\int_{a}^{b}(f-\lambda g)^{2}\) for some \(\lambda \in \mathbf{R}\) and \(0<\) \(\int_{a}^{b}(f-\lambda g)^{2}\) for all \(\lambda \in \mathbf{R}\). (b) If equality holds, must \(f=\lambda g\) for some \(\lambda \in \mathbf{R} ?\) What if \(f\) and \(g\) are continuous?

Step-by-Step Solution

Answer

Proving \( \left| \int_{a}^{b} f g \right| \leq \left( \int_{a}^{b} f^{2} dx \right)^{\frac{1}{2}} \cdot \left( \int_{a}^{b} g^{2} dx \right)^{\frac{1}{2}} \) involves expanding a squared integral and solving the resulting quadratic inequality. If equality holds, then \( f = \lambda g \) almost everywhere, and if \( f \) and \( g \) are continuous, \( f = \lambda g \) for all \( x \) in \( [a, b] \).

Step 1 - Squaring a Difference

Consider the integral \( \int_{a}^{b} (f - \lambda g)^{2} dx \) for some \( \lambda \) in \( \mathbf{R} \).

Step 2 - Expanding the Square

Expand the square \( \int_{a}^{b} (f - \lambda g)^{2} dx \) to get \( \int_{a}^{b} (f^{2} - 2\lambda fg + \lambda^{2} g^{2}) dx \).

Step 3 - Splitting the Integral

Split the expanded integral: \( \int_{a}^{b} f^{2} dx - 2\lambda \int_{a}^{b} fg dx + \lambda^{2} \int_{a}^{b} g^{2} dx \).

Step 4 - Defining Notation

Define \( I = \int_{a}^{b} fg dx \). The integral becomes: \( \int_{a}^{b} f^{2} dx - 2\lambda I + \lambda^{2} \int_{a}^{b} g^{2} dx \).

Step 5 - Setting Integral to Zero

Setting the integral to zero: \( \int_{a}^{b} f^{2} dx - 2\lambda I + \lambda^{2} \int_{a}^{b} g^{2} dx = 0 \).

Step 6 - Solving Quadratic Equation

Consider it as a quadratic equation in \( \lambda \): \( A - 2\lambda I + \lambda^{2} B = 0 \), where \( A = \int_{a}^{b} f^{2} dx \) and \( B = \int_{a}^{b} g^{2} dx \).

Step 7 - Solving for Lambda

For the discriminant \( \Delta \) of the quadratic equation to be non-negative: \( \Delta = 4I^{2} - 4AB \). Since the square of the integral must be zero or positive, set \( I^{2} \leq AB \).

Step 8 - Taking the Square Root

Taking the square root of both sides: \( |I| \leq \sqrt{A} \cdot \sqrt{B} \). Substituting back the variables: \( \left| \int_{a}^{b} f g \right| \leq \left( \int_{a}^{b} f^{2} dx \right)^{\frac{1}{2}} \cdot \left( \int_{a}^{b} g^{2} dx \right)^{\frac{1}{2}} \).

Step 9: Part b Step 1 - Equality Condition

If the equality holds, and since the discriminant must be zero, we have \( \int_{a}^{b} (f - \lambda g)^{2} = 0 \), implying \( f = \lambda g \) almost everywhere.

Step 10: Part b Step 2 - Continuous Functions

If both \( f \) and \( g \) are continuous, then equality means \( f \) and \( g \) must be proportional everywhere in \( [a, b] \).

Key Concepts

Integration

Integration is a fundamental concept in calculus and is used to find the area under a curve. When we encounter exercises related to the Cauchy-Schwarz inequality, integration helps us evaluate the relationships between functions over an interval. In this exercise, we are given two functions, \( f \) and \( g \), that are defined on the interval \( [a, b] \). We need to integrate the product, squares, and differences of these functions to prove the inequality.
Specifically, we evaluated the integral of the square of the difference \( (f - \lambda g)^2 \), expanded and split it into separate integrals. Make sure to fully understand the integral properties handled in each step to derive the desired inequality effectively.

Quadratic Equations

Quadratic equations are equations of the form \( ax^2 + bx + c = 0 \). In the context of this exercise, once we expanded the integral of \( (f - \lambda g)^2 \), it resulted in a quadratic in \( \lambda \). We expressed this as \( A - 2\lambda I + \lambda^{2} B = 0 \) with \( A \) and \( B \) being the integrals of squared functions, and \( I \) as the integral of their product.
Solving the quadratic equation involved using the discriminant concept \((b^2 - 4ac)\) to figure out conditions for which this equation holds true or reaches zero. Here, setting the discriminant \( 4I^2 - 4AB \) to be non-negative enabled us to conclude that \( I^2 \leq AB \), leading to our desired result.

Proportional Functions

Proportional functions are functions that are scalar multiples of one another, represented as \( f = \lambda g \) where \( \lambda \) is a constant. In this problem, understanding proportional functions is crucial for part b of the exercise.
If the equality condition holds in Cauchy-Schwarz inequality, it means the discriminant is zero, i.e., \( \int_{a}^{b} (f - \lambda g)^2 = 0 \). Since the integral of a squared function can only be zero if the function itself is zero almost everywhere, it implies that \( f \) must be some constant multiple of \( g \).
This relationship becomes even more precise if \( f \) and \( g \) are continuous. In such cases, \( f \) and \( g \) must be proportional at every point in the interval \([a, b]\). This continuity ensures that the proportional relationship holds consistently across the domain.

Continuous Functions

Continuous functions are functions that have no breaks, holes, or jumps in their domains. Mathematically, a function \( f \) is continuous at a point \( x = c \) if \( f(c) \) is defined and \( \lim\limits_{x \to c} f(x) = f(c) \). For this exercise, the continuous nature of \( f \) and \( g \) plays a significant role in proving the equality condition in part b.
When both \( f \) and \( g \) are continuous over \( [a, b] \), any relationship derived from the integral equations holds true across every point in that interval. This means if the Cauchy-Schwarz equality holds, \( f \) and \( g \) must be proportional everywhere within \( [a, b] \), rather than just almost everywhere. Thus, the continuity ensures a stronger, more definitive form of the result, making this concept key to understanding the precision of the equality condition.