Question

Show that if \(f\) and \(g\) are bounded, \(K\)-Lipschitz functions from the normed linear space \(X\) to \(\mathbb{R}\) then \(f . g\) is Lipschitz. Give a counterexample when \(f\) and \(g\) are unbounded.

Short answer

In summary, we have proven that if f and g are bounded, K-Lipschitz functions from the normed linear space X to ℝ, then their product fg is Lipschitz with the Lipschitz constant 2MK. We have also shown a counterexample in which unbounded functions f and g, although individually Lipschitz, have a product fg that is not Lipschitz.

Step-by-step solution

Definition of K-Lipschitz functions

Recall that if f and g are K-Lipschitz functions, then for every \(x, y \in X\), we have: $$|f(x) - f(y)| \leq K\cdot||x - y||$$ and $$|g(x) - g(y)| \leq K\cdot||x - y||$$

Using the K-Lipschitz property of f and g to derive the Lipschitz condition for fg

Consider the product function fg, and let's derive the Lipschitz condition for it. We want to find an upper bound for \(|f(x)g(x) - f(y)g(y)|\). Using the product-sum identity, we can rewrite the expression as follows: $$|f(x)g(x) - f(y)g(y)| = |f(x)g(x) - f(x)g(y) + f(x)g(y) - f(y)g(y)|$$ Now, by the triangle inequality, we have: $$|f(x)g(x) - f(x)g(y) + f(x)g(y) - f(y)g(y)| \leq |f(x)g(x) - f(x)g(y)| + |f(x)g(y) - f(y)g(y)|$$ Applying the K-Lipschitz property of f and g, we get: $$|f(x)g(x) - f(x)g(y)| \leq |f(x)||g(x) - g(y)| \leq M\cdot K\cdot||x - y||$$ and $$|f(x)g(y) - f(y)g(y)| \leq |g(y)||f(x) - f(y)| \leq M\cdot K\cdot||x - y||$$ Where M is the supremum of the maximum values of the functions f and g. Adding these inequalities: $$|f(x)g(x) - f(y)g(y)| \leq M\cdot K\cdot||x - y|| + M\cdot K\cdot||x - y|| = 2M\cdot K\cdot||x - y||$$ Thus, we have shown that the product function fg is Lipschitz with Lipschitz constant 2MK. #Part 2: Give a counterexample when f and g are unbounded#

Construct unbounded functions

Consider the following unbounded functions: $$f(x) = x \quad \text{and} \quad g(x) = \frac{1}{x}, \quad \text{for} \quad x \neq 0$$ $$f(0) = 0 \quad \text{and}\quad g(0) = 0$$ Clearly, both f and g have no upper bound. However, both f and g are Lipschitz functions as: $$|f(x) - f(y)| = |x - y| \leq 1\cdot ||x - y||$$ and $$|g(x) - g(y)| = \left|\frac{1}{x} - \frac{1}{y}\right| = \frac{|x - y|}{|x|\cdot|y|} \leq 1\cdot||x - y||$$

Show that fg is not Lipschitz

Consider the product function fg: $$fg(x) = x\cdot\frac{1}{x} = 1, \quad \text{for} \quad x \neq 0$$ $$fg(0) = 0$$ Now, let's examine the limit of \(\frac{|fg(x) - fg(y)|}{||x - y||}\) as \(x \to y\). $$\lim_{x \to y} \frac{| fg(x) - fg(y) |}{||x - y||} = \lim_{x \to y} \frac{| fg(x) - fg(0) |}{||x - 0||} = \lim_{x \to 0} \frac{|1 - 0|}{|x|} = \lim_{x \to 0} \frac{1}{|x|}$$ This limit does not exist because it goes to infinity, which means that the function fg is not Lipschitz. We have now successfully found a pair of unbounded functions f and g whose product is not Lipschitz.

Key concepts

Normed Linear Space

When we talk about the setting for Lipschitz functions, we often begin with the concept of a normed linear space. Imagine a space where you can add elements together and multiply them by scalars — that's the linear part. The 'normed' bit comes from the ability to measure 'size' or 'length' of elements in this space. The norm is represented with double bars like this: \( ||x|| \) for any element \( x \) in our space. It must satisfy certain properties like being non-negative and only zero if the element is the zero of the space. It should also behave well under scalar multiplication and satisfy the triangle inequality, which helps us to navigate through the space intuitively as well.

Understanding the structure of normed linear spaces is crucial because it's the framework where the behavior of Lipschitz functions is analyzed and where the concept of distance is clearly defined, which is key to understanding Lipschitz continuity.

Bounded Functions

Moving on to understand what it means for a function to be bounded in this context, think of it as having a 'fence' around its possible values. For every input in their domain, bounded functions return outputs that don’t go beyond a certain fixed boundary. Mathematicians formally say that a function \( f \) is bounded if there exists a real number \( M \) such that for all \( x \) in the domain of \( f \) the value of \( |f(x)| \) is less than or equal to \( M \) .

This concept is important for proving certain properties of functions, like in the exercise where the boundedness of \( f \) and \( g \) ensures the bounded behavior of their product, even when involving distances measured within the space.

Triangle Inequality

The triangle inequality is a fundamental rule in geometry that also plays a protagonist role in normed linear spaces. If you have two vectors \( x \) and \( y \) in this space, moving from the origin to \( x \) and then to \( y \) should be at least as long as going directly from the origin to \( y \) . Mathematically, it's written as \( ||x + y|| \leq ||x|| + ||y|| \) for any vectors \( x \) and \( y \) .

In the context of Lipschitz functions, the triangle inequality helps to manage and estimate the 'distance' between function values based on the inputs. It's a stepping stone for establishing the Lipschitz condition of a function, as seen when separating the product of two functions into parts.

Lipschitz Constant

Finally, let's talk about the Lipschitz constant. It's a real number designated by \( K \) that essentially describes 'how fast' a Lipschitz function can change. Put mathematically, for a \( K \) -Lipschitz function \( f \) , you have the inequality \( |f(x) - f(y)| \leq K \cdot\| x - y \|| \) for all pairs \( x, y \) from the space. If there's a \( K \) making this true, the function \( f \) is under control, never swinging wildly for close inputs — it's predictably tame. The Lipschitz constant acts like a leash on the function's rate of change.

In the exercise, using the Lipschitz constant \( K \) , we were able to constrain the product of two bounded functions and show it's also Lipschitz under the right conditions. This kind of analysis is invaluable when dealing with the stability and convergence properties of functions.