Question

Prove: If \(\left\\{F_{n}\right\\}\) converges uniformly to \(F\) on \(S,\) then \(\lim _{n \rightarrow \infty}\left\|F_{n}\right\|_{S}=\|F\|_{S}\).

Short answer

Question: Prove that if a sequence of functions \(\left\\{F_{n}\right\\}\) converges uniformly to a function \(F\) on a set \(S,\) then the limit of their respective norms is equal to the norm of the limiting function. Solution: Given that \(\left\\{F_{n}\right\\}\) converges uniformly to \(F\) on \(S,\) we showed that $$\lim _{n \rightarrow \infty}\left(\left\|F_{n}\right\|_{S} - \|F\|_{S}\right) = 0$$ by using the definitions of uniform convergence and function norms, as well as the triangle inequality for function norms. Thus, we concluded that $$\lim _{n \rightarrow \infty}\left\|F_{n}\right\|_{S}=\|F\|_{S},$$ and the statement is proved.

Step-by-step solution

Definition of Uniform Convergence

In order for a sequence of functions \(\left\\{F_{n}\right\\}\) to be said to uniformly converge to a function \(F\) on a set \(S,\) it must satisfy the following condition: For any given real number \(\varepsilon > 0,\) there exists a natural number \(N\) such that for any \(n \geq N\) and any \(x \in S,\) we have $$|F_n(x) - F(x)| < \varepsilon.$$

Definition of Function Norm

For a function \(F\) defined on a set \(S,\) its norm, denoted as \(\|F\|_{S},\) is given by $$\|F\|_{S} = \sup_{x \in S} |F(x)|.$$ It represents the maximum absolute value of \(F(x)\) over the set \(S.\)

Show the difference between norms is less than or equal to the difference between functions

We want to show that $$\lim _{n \rightarrow \infty}\left\|F_{n}\right\|_{S}=\|F\|_{S}.$$ We can manipulate this expression by subtracting \(\|F\|_{S}\) from both sides. So, this is equivalent to proving that $$\lim _{n \rightarrow \infty}\left(\left\|F_{n}\right\|_{S} - \|F\|_{S}\right) = 0.$$ Now, notice that $$\left\|F_{n}\right\|_{S} - \|F\|_{S} \leq \|F_n - F\|_{S},$$ which follows from the triangle inequality for function norms.

Use uniform convergence of the sequence of functions

Since \(\left\\{F_{n}\right\\}\) converges uniformly to \(F\) on \(S,\) for any given real number \(\varepsilon > 0,\) there exists a natural number \(N\) such that for any \(n \geq N\) and any \(x \in S,\) we have $$|F_n(x) - F(x)| < \varepsilon.$$ Taking the supremum over \(S,\) we get $$\|F_n- F\|_{S} < \varepsilon,$$ for all \(n \geq N\).

Combine the results from Step 1 and Step 2 and conclude the proof

From Steps 1 and 2, we have that $$\left\|F_{n}\right\|_{S} - \|F\|_{S} \leq \|F_n - F\|_{S} < \varepsilon,$$ for all \(n \geq N.\) As this inequality holds for any arbitrary \(\varepsilon > 0,\) we conclude that $$\lim _{n \rightarrow \infty}\left(\left\|F_{n}\right\|_{S} - \|F\|_{S}\right) = 0,$$ which implies that $$\lim _{n \rightarrow \infty}\left\|F_{n}\right\|_{S}=\|F\|_{S}.$$ Thus, the exercise is proved.

Key concepts

Real Analysis

Real analysis is a branch of mathematics that deals with the study of real numbers and real-valued functions. It involves the rigorous analysis of the concepts of limit, continuity, differentiation, and integration. At the heart of real analysis is understanding the behavior of sequences and series of functions, and exploring the convergence properties of these objects.

In our case, we are interested in the convergence of a sequence of functions to a specific function. This leads us to the discussion of uniform convergence, which plays a crucial role in ensuring that many important properties of functions are preserved as the sequence converges. A rich understanding of real analysis is essential to grasp advanced mathematical concepts and to pursue scientific research or studies in fields that require mathematical rigor.

Function Norm

The notion of a function norm is a pivotal concept in real analysis, serving as a measure of the size of a function. The norm of a function can be thought of in a similar way to how we consider the length of a vector in vector space.

For a given real-valued function defined on a set, the norm, denoted as \( \|F\|_{S} \), is the supremum of the absolute values of the function over the set. This effectively captures the 'largest' value the function takes and gives a uniform measure of its magnitude across the domain. As such, the function norm plays an instrumental role in evaluating the convergence of function sequences, particularly in the context of uniformly convergent functions, by providing a clear benchmark for comparing the magnitudes of different functions.

Supremum of a Function

The supremum, or 'sup' for short, is a term from the realm of real analysis that refers to the least upper bound of a set. When applied to a function, the supremum is the smallest real number that is greater than or equal to the function at every point in its domain. For real-valued functions, finding the supremum is akin to identifying the peak value the function can reach, without necessarily being a value the function actually attains.

In the context of a function norm, the supremum is used to define the maximum magnitude of the function over its domain and is represented as \(\sup_{x \in S} |F(x)|\). Calculating the supremum of a function can sometimes be challenging, as it requires a comprehensive understanding of the function's behavior throughout its entire domain.

Sequence of Functions

A sequence of functions is a collection of functions in which each function is assigned to a distinct natural number in a specific order, forming a countable set. In real analysis, the convergence of such sequences is of significant interest, especially when dealing with infinite processes or approximations.

A sequence of functions \(\left\{F_{n}\right\}\) may converge pointwise or uniformly to some limiting function \(F\) as \(n\) approaches infinity. Pointwise convergence refers to convergence at each individual point in the domain, while uniform convergence implies that the functions converge together as a whole at a uniform rate over the entire domain. Uniform convergence is crucial for the preservation of continuity, integrability, and differentiability of the limit function, hence its importance in various mathematical analyses and applications.