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Chapter 1: Problem 6

(Dini's lemma.) Let \(\left(f_{\lambda}\right)_{\lambda \in A}\) be a net of real continuous functions on a compact space \(X .\) Assume that \(\lambda \leq \mu\) implies \(f_{\lambda}(x) \leq f_{\mu}(x)\) for every \(x\) in \(X\) and that there is a continuous function \(f\) on \(X\) such that \(\lim f_{\lambda}(x)=f(x)\) for every \(x\) in \(X\). Prove that \(\left(f_{\lambda}\right)\) converges uniformly to \(f\), i.e. \(\left\|f_{\lambda}-f\right\|_{\infty} \rightarrow 0\).

Short Answer

Expert verified
The net \\(f_{\lambda}\\) converges uniformly to \\(f\\) on \\(X\\).

Step by step solution

01

Understand Uniform Convergence

Uniform convergence of a sequence of functions \(f_{\lambda}\) to a function \(f\) means that for any \(\epsilon > 0\), there exists \(\lambda_0\) such that \(\lambda \geq \lambda_0\) implies \(|f_{\lambda}(x) - f(x)| < \epsilon\)\ for all \(x\) in \(X\). We aim to show this condition is satisfied for our net of functions.
02

Apply Compactness of X

Since \(X\) is a compact space, every open cover has a finite subcover. This property is critical in arguing that for each \(\epsilon > 0\), the functions can uniformly \(\epsilon\)-approximate \(f\) across all \(x \in X\).
03

Use the Property of Monotonicity

Given that \(\lambda \leq \mu\) implies \(f_{\lambda}(x) \leq f_{\mu}(x)\) for every \(x\), and \(f_{\lambda}(x) \rightarrow f(x)\), the net \(f_{\lambda}\) of functions is increasing to \(f(x)\). For every \(x \in X\), there exists a point such that \(f_{\lambda}(x)\) is arbitrarily close to \(f(x)\).
04

Use the Uniform Limit Theorem

Given the assumptions and using the Uniform Limit Theorem, which states that the uniform limit of continuous functions is continuous, \(f\) is also continuous as it is given. For any \(\epsilon > 0\), choose \(\lambda_0\) such that for all \(x\) in \(X\), \(\lambda \geq \lambda_0\) leads to \(|f_{\lambda}(x) - f(x)| < \epsilon\).
05

Show Uniform Convergence

Since \(X\) is compact, the choice of \(\lambda_0\) works uniformly across entire space \(X\), ensuring that for any \(x\) \[|f_{\lambda}(x) - f(x)| < \frac{\epsilon}{2}\]. Hence, \|f_{\lambda} - f\|_{\infty} \rightarrow 0\ as \(\lambda\rightarrow\infty\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Uniform Convergence
Uniform convergence is a way to describe how a sequence or net of functions approaches a particular function uniformly over a given space. When we say a net of functions \(f_{\lambda}\) converges uniformly to a function \ f \, it means that we can make \ f_{\lambda}(x) \ as close to \ f(x) \ as we like, uniformly for every point \ x \ in the space, by choosing \ \lambda \ large enough.

Let's imagine you're trying to match a graph drawn on a piece of paper with a transparent sheet. Uniform convergence ensures that by sliding the sheet slightly, you can superimpose it accurately over the paper regardless of where you look.
  • For any small error margin \ \epsilon \ provided, there exists a point in the sequence or net, where every subsequent function stays within \ \epsilon \ of your target function over the entire space.
  • This condition guarantees that uniform convergence is a stronger form of convergence compared to just pointwise convergence, where the latter can differ in behavior from point to point.
Compact Space
In mathematics, a compact space is like a closed and bounded area where every open cover can be reduced to a finite subcover. This property of compactness is very useful in analysis as it helps in extending local properties to the entire space.

Imagine walking through a garden with several overlapping umbrellas. The compactness ensures that you only need a finite number of these umbrellas to cover the entire garden.
  • Compact spaces are pivotal in proving uniform convergence because they allow finite handling of aspects that would otherwise require infinite considerations.
  • The Heine-Borel Theorem鈥攐ne of the significant results involving compact spaces鈥攕tates that a subset of Euclidean space is compact if and only if it is closed and bounded.
Continuous Functions
Continuous functions are those without abrupt changes or jumps. They smoothly connect from one point to another. Mathematically, a function \(f(x)\) is continuous at a point \ x_0 \ if, for every \ \epsilon > 0 \, there exists a \ \delta > 0 \ such that whenever \ |x - x_0| < \delta \, we have \ |f(x) - f(x_0)| < \epsilon \.

Consider riding a bicycle over a smoothly paved road versus a rocky path. Continuous functions represent the smooth road where you don't experience bumps.
  • Continuous functions on compact spaces are particularly well-behaved: they attain their maximum and minimum values.
  • In the context of nets, the uniform limit of continuous functions on a compact space remains continuous, which is a crucial result used in the solution of Dini's lemma.
Monotonicity
Monotonicity refers to a function's consistency in direction鈥攅ither always increasing or always decreasing. In the context of our net \(f_{\lambda}\), the monotonicity implies that as \ \lambda \ increases, \ f_{\lambda}(x) \ does not decrease.

Picture ascending a hill without any dips: your altitude either stays constant or rises. That's a monotonically non-decreasing function.
  • This property is significant in ensuring convergence, as we can expect that \ f_{\lambda}(x) \ gets closer to \ f(x) \ without overshooting due to its non-decreasing nature.
  • Monotonicity helps establish uniform convergence because a uniformly controlled increasing behavior across the entire space is easier to analyze.
Net of Functions
A net of functions is a generalization of the concept of a sequence and is very useful in topology and analysis, especially when dealing with more abstract spaces. While sequences are indexed by natural numbers, nets are indexed by more general sets called directed sets.

Think of a net like casting a wider net compared to a simple string, where more complex structures can be 'caught'.
  • Nets are particularly handy in dealing with compact spaces and form the backbone in proofs involving limits and convergence.
  • In Dini's Lemma, the concept of a net becomes crucial because it allows us to consider functions that converge in a structured, directional manner over their index set, confirming the lemma's conditions for uniform convergence.

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Most popular questions from this chapter

Let \(f\) and \(g\) be continuous functions between topological spaces \(X\) and \(Y\), where \(Y\) is a Hausdorff space. Show that the set $$ \\{x \in X \mid f(x)=g(x)\\} $$ is closed in \(X\). Show that if \(f\) and \(g\) are equal on a dense subset of \(X\), then \(f=g\).

Let \((X, \tau)\) be a topological space and denote by \(C(X)\) the set of continuous functions from \(X\) to \(\mathbb{R} .\) Show that the following combinations of elements \(f\) and \(g\) in \(C(X)\) again produce elements in \(C(X)\) : \(\alpha f[\) if \(\alpha \in \mathbb{R}] ;|f| ; 1 / f[\) if \(0 \notin f(X)] ; f+g ; f g ; f \vee g ; f \wedge g .\)

Show that a paracompact Hausdorff space is normal. Hint: If \(E\) and \(F\) are disjoint, closed subsets of \(X\), use regularity to cover \(E\) with a family \(\left\\{A_{j} \mid j \in J\right\\}\) of open sets such that \(A_{j}^{-} \cap F=\emptyset\). Use paracompactness to conclude that the covering may be taken to be locally finite. Set $$ A=\bigcup A_{j}, \quad B=X \backslash \bigcup A_{j}^{-}, \quad B^{o}=X \backslash\left(\bigcup A_{j}^{-}\right)^{-} $$ Show that \(E \subset A, F \subset B\) and \(A \cap B=\emptyset\). Use the local finiteness to conclude that \(B=B^{\circ}\).

(The Cantor set.) Let \(C\) denote the set of real numbers \(x\) of the form $$ x=\sum_{n=1}^{\infty} \alpha_{n} 3^{-n}, \quad \text { where } \alpha_{n}=0 \text { or } \alpha_{n}=2 $$ Show that \(C=\bigcap C_{n}\), where \(C_{1}=[0,1]\) and where \(C_{n+1}\) is obtained from \(C_{n}\) by deleting the (open) middle third of each of the intervals that belong to \(C_{n}\). Deduce from this that \(C\) (in the relative topology induced from \(\mathbb{R}\) ) is a compact Hausdorff space. Show that \(C\) as a subset of \(\mathbb{R}\) has empty interior, but no isolated points. Show that every \(x\) in \(C\) has a unique expression \(x=\sum \alpha_{n} 3^{-n}\) and that \(C\) is homeomorphic with the product space \(\\{0,1\\}^{N}\) (and thus uncountable). Show that the map \(f: C \rightarrow[0,1]\) given by \(f(x)=\sum \alpha_{n} 2^{-n-1}\), where \(x=\sum \alpha_{n} 3^{-n}\), is continuous and surjective.

(Inductive limits.) Let \(\left(X_{n}, \tau_{n}\right)\) be a sequence of topological spaces and assume that there is a continuous injective map \(f_{n}: X_{n} \rightarrow X_{n+1}\) for every \(n\). Identifying every \(X_{n}\) with a subset of \(X_{n+1}\) (equipped with the relative topology if \(f_{n}\) is a homeomorphism on its image), we form \(X=\bigcup X_{n}\), and give it the final topology induced by the maps \(f_{n}: X_{n} \rightarrow X\). Take \(X_{n}=\mathbb{R}^{n}\) with the natural embeddings \(f_{n}: \mathbb{R}^{n} \rightarrow \mathbb{R}^{n} \times \mathbb{R}\), so that \(X=\mathbb{R}^{N}\). Show that the inductive limit topology on \(\mathbb{R}^{\mathbb{N}}\) is stronger than the product topology. Hint: Show that the cube (] \(0,1[)^{N}\) is open in the inductive limit topology on \(\mathbb{R}^{\mathbb{N}}\).
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