Question

Prove the Cauchy-Schwarz Inequality for Integrals: $$\left[\int_{a}^{b} f(x) g(x) d x\right]^{2} \leq \int_{a}^{b} f^{2}(x) d x \int_{a}^{b} g^{2}(x) d x$$ Hint: Consider the double integral of $$F(x, y)=[f(x) g(y)-f(y) g(x)]^{2}$$ over the rectangle \(R=\\{(x, y): a \leq x \leq b, a \leq y \leq b\\}\).

Short answer

The integral of the squared difference implies Cauchy-Schwarz due to non-negative integrals.

Step-by-step solution

Understand the Problem Setup

We have the functions \(f(x)\) and \(g(x)\) over the interval \([a, b]\), and we are tasked with proving that the Cauchy-Schwarz inequality holds for their integrals. The inequality is given as: \(\left[\int_{a}^{b} f(x) g(x) d x\right]^{2} \leq \int_{a}^{b} f^{2}(x) d x \int_{a}^{b} g^{2}(x) d x\). We also have the function \(F(x, y) = [f(x) g(y) - f(y) g(x)]^2\), which needs to be integrated over the rectangle \(R\).

Consider the Properties of \( F(x, y) \)

Notice that \( F(x, y) \geq 0 \) for all \(x, y\) in the rectangle \(R = \{(x, y): a \leq x \leq b, a \leq y \leq b\}\). This non-negativity is crucial as it implies that the integral of \( F(x, y) \) over \(R\) is also non-negative.

Expand the Function \( F(x, y) \)

We expand \( F(x, y) = [f(x)g(y) - f(y)g(x)]^2 = f^2(x)g^2(y) - 2f(x)g(y)f(y)g(x) + f^2(y)g^2(x) \). This expansion results in three components: cross terms and squared terms.

Integrate \( F(x, y) \) over \( R \)

Integrate the expression for \( F(x, y) \) over \(R\). We have:1. \( \int_{a}^{b} \int_{a}^{b} f^2(x)g^2(y) \, dx \, dy \).2. \( \int_{a}^{b} \int_{a}^{b} f(x)g(y)f(y)g(x) \, dx \, dy \).3. \( \int_{a}^{b} \int_{a}^{b} f^2(y)g^2(x) \, dx \, dy \).Combining these, the overall integral of \( F(x, y) \) is non-negative.

Apply the Non-Negativity Conclusion

Since \( \int_{R} F(x, y) \, dx \, dy \geq 0 \), this implies:\[ \int_{a}^{b} f^2(x) \, dx \int_{a}^{b} g^2(y) \, dy \geq \left( \int_{a}^{b} f(x) g(x) \, dx \right)^2 \]Moreover, this elucidates the integration of inner products and validates the Cauchy-Schwarz inequality across \([a, b]\).

Interpret the Result

This inequality describes a fundamental property of integrals over vectors (functions seen as vectors in function space), relating to how their normed magnitudes compare when aligned versus independently squared.

Key concepts

Integral Inequalities

Integral inequalities, such as the Cauchy-Schwarz Inequality, play a pivotal role in mathematical analysis. They help understand relationships between integrals of functions over certain intervals. This particular inequality states that the square of an integral of the product of two functions is less than or equal to the product of their individual squared integrals.

This form of inequality can often reveal deeper insights about the behavior and interaction of the functions involved. In the case of the Cauchy-Schwarz Inequality, it provides a boundary on how much two functions can "overlap" when viewed as geometric objects (such as vectors) in a function space.Q4AAD6

Integral inequalities are foundational in various fields of mathematics and physics, allowing for estimation and constraint of function behavior in complicated systems.

Functions

Functions are the core building blocks in calculus and mathematical analysis. They represent relationships between sets, typically associating each element of one set with an element of another. In our exercise, we deal with two functions, \(f(x)\) and \(g(x)\), defined over a specific interval [a, b].

These functions can be seen as vectors if we consider each point in their domain to be a component. The integral of a function over an interval can be thought of as the "sum" of all these components, providing a measure for the function's behavior over the entire interval.

In this exercise, we aim to understand how these functions interact through their product, represented in the Cauchy-Schwarz Inequality. This interaction allows us to study their combined effect, offering insights into their geometric and algebraic properties.

Mathematical Proofs

Mathematical proofs form the backbone of mathematical reasoning and validation. They offer a way to substantiate and prove hypotheses or theories by logical deduction. In the context of this exercise, we are tasked with proving the Cauchy-Schwarz Inequality for integrals.

The process involves setting up the problem by defining the necessary functions and understanding their domain. The core of the proof relies on examining a constructed function, \(F(x, y)\), which exhibits certain symmetric properties. This function's non-negativity is central to establishing the inequality, as it serves as a bridge to link the integrals involved.

Breaking down \(F(x, y)\) into simpler components allows us to confirm its non-negativity by integrating over the specified region \(R\). Each of these steps contributes to concluding that the original inequality holds independent of the specific functions chosen.

Integral Calculus

Integral calculus is a fundamental aspect of calculus, focusing on the accumulation of quantities. It involves concepts such as integration, which generalizes the process of summing data points to find the whole's magnitude or value.

In our exercise, we're using integral calculus to explore the relationship between two functions via the Cauchy-Schwarz Inequality. The calculations involve integrating not only the functions themselves but also expressions involving their products over a two-dimensional region.

The double integral technique used in the solution helps us understand the interactions between the functions across a rectangle \(R\). It provides a multidimensional perspective on integration, essential for dealing with complex relationships in mathematics and physics. These insights gained reinforce the importance of integral calculus as a tool for solving real-world problems.