Real analysis theorems

So in this post I’ll summarize a bunch of theorems whose proofs we had to know for the final exam, along with the general idea of the proof. This doesn’t even include all the stuff we had to know from the assignments, either… too much stuff to memorize >.< going crazy right now

  • Uniform limit of continuous functions is continuous. Proof: Use uniform convergence to find appropriate f_N which has distance less than \epsilon/3 away from the limit. Find \delta > 0 from the continuity of f_N. Apply triangle inequality trick with this particular f_N.
  • C_b(X) is complete. Proof: Pick any Cauchy sequence. Examine the pointwise sequences; these are Cauchy in \mathbb{R} (by completeness) and hence converge. Define f to be the pointwise limit. Examine |f_n(x) - f_m(x)| as m \to \infty to prove that indeed, the convergence is uniform. Hence the limit is continuous, and a simple argument also shows it’s bounded, so it lives in C_b(X).
  • [Weierstrass M-Test] A normed linear space (X,\| \cdot \|) is complete if and only if whenever \{ x_n \} is a sequence with \sum_{n=1}^\infty \| x_n \| < \infty, we have \sum_{n=1}^\infty x_n converges. Proof: The partial sums of the norm series converge and are hence Cauchy. The triangle inequality then implies that the partial sums of the series itself are Cauchy, so by completeness, we get convergence.
  • [Baire Category] If (X,d) is a complete metric space and \{ U_n \}_{n=1}^\infty is a sequence of open dense sets in X, then their intersection is dense. Proof: We prove that any open set in X contains an element of \bigcap U_n. Choose x_1 \in X and 0 < r_1 < 1 with B(x_1,r_1) \subseteq B[x_1,r_1] \subseteq W \cap U_1. Now choose x_2 \in X and 0 < r_2 < 1/2 and x_2 \in B[x_1,r_1] \cap U_2. And so on. Get a sequence B[x_1, r_1] \supseteq B[x_2,r_2] \supseteq \ldots with r_n \to 0 and so Cantor’s Intersection Theorem.
  • [Banach Contractive Mapping] If (X,d) is complete and \Gamma : X \to X is contractive of index 0 < k < 1 then there is unique x_0 \in X with \Gamma(x_0) = x_0. Proof: For uniqueness, note that if \Gamma(x_0) = x_0 and \Gamma(y_0) = y_0 then d(x_0, y_0) = d(\Gamma(x_0), \Gamma(y_0)) \leq k \cdot d(x_0, y_0) means d(x_0,y_0) = 0. For existence, just pick an arbitrary point and iterate \Gamma over and over to get a sequence. Apply triangle inequality to obtain Cauchyness, and hence convergence. Continuity of \Gamma gives that the limit is a fixed point since \Gamma(x_n) \to \Gamma(x_0) and \Gamma(x_n) = x_{n+1} \to x_0.
  • [Arzela-Ascoli (one direction)] Let (X,d) be compact. If \mathcal{F} \subseteq C(X) then it is relatively compact if and only if it is pointwise bounded and equicontinuous. Proof (forward): Relatively compact implies totally bounded, so get a finite \tfrac{\epsilon}{3}-mesh for \mathcal{F}. X is compact, so continuity is always uniform. Exploit the uniform continuity of the stuff in the mesh, and then exploit the fact that it’s a mesh. Triangle inequality trick: the proof is basically a replica of the “uniform limit of continuous functions” theorem. We get uniform boundedness since the stuff in the mesh is bounded (and it’s a mesh).
  • [Weierstrass Approximation] If f \in (C[a,b], \| \cdot \|_{\infty}) then there is a sequence of polynomials \{ p_n \} converging uniformly to f on [a,b]. Proof: Can’t really summarize this nicely at the moment. Not exactly my favourite proof -_-.
  • [Stone-Weierstrass (Lattice Version)] If (X,d) is compact and we have a point-separating subspace \Phi \subseteq C(X) which contains the function 1 and also contains \max\{ f, g \} for any two functions f, g then \Phi is dense in C(X).
  • [Stone-Weierstrass (Subalgebra Version)] If (X,d) is compact and we have a point-separating unital subalgebra \Phi \subseteq C(X) then it is dense in C(X). Proof: Can assume A is closed. Invoke Weierstrass Approximation Theorem to find a sequence of polynomials converging uniformly to the absolute value function on [-M, M] where M is a bound for f \in A. Hence by considering p_n \circ f, prove that |f| \in A. Thereafter, prove that \max\{ f, g \} = \frac{1}{2}{f(x) + g(x) + |f(x) - g(x)|} \in A and invoke Lattice Version.
  • \mathbb{Q} is not G_\delta. Proof: Assume it is. By manipulating some intersections and unions, we quickly derive that \mathbb{R} is first category, contradicting Baire Category.
  • (X,d) is compact if and only if every family of closed sets with FIP has nonempty intersection. Proof: Since each F_n is closed, F_n^c is open. So if the intersection is empty, then by taking its complement, we actually get an open cover of X! Now we can take a finite subcover, complement again and hence contradict FIP. Other direction is equally simple.
  • Continuous image of compact set is compact. Proof: If we have an open cover of f(X) we can pull each set back, and by continuity of f the pullbacks are all open, yielding an open cover of X. Pick finite subcover, push it the other way and you’re done.
  • Continuity on a compact set is always uniform. Proof: Let \epsilon > 0 and for all x \in X pick \delta_x. Then write X = \bigcup_{x \in X} (x-\delta_x, x+\delta_x). Pick a finite subcover and then let \delta be the minimum of all those appearing in this subcover. Done.
  • If (X,d) is compact then every infinite set has a cluster point. Proof: Just pick a sequence from this set with all the terms distinct. By compactness, it has a convergent subsequence and hence a cluster point, by distinctness.
  • If (X,d) is sequentially compact then it is totally bounded. Proof: If we assume it is not totally bounded then there exists an \epsilon such that no \epsilon-mesh exists, so we can pick a sequence whose terms are always at least \epsilon apart. This will (obviously) not have a convergent subsequence, contradicting sequential compactness.
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About mlbaker

just another guy trying to make the diagrams commute.
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4 Responses to Real analysis theorems

  1. Wow, I only understood # 1 and # 3. I have a lot of catching up to do in MATH247 before I start PMATH351.

    The other proofs do look very interesting though.

  2. matto says:

    in the 3rd last one “Continuity on a compact set is always uniform” you have an interval where you should have a ball

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