Tips and Tricks in Real Analysis

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1 Tips and Tricks in Real Analysis Nate Eldredge August 3, 2008 This is a list of tricks and standard approaches that are often helpful when solving qual-type problems in real analysis. Approximate. There are a lot of results that say that a function f can be approximated by a sequence of nicer functions f n, so that f n f in some appropriate sense. So if you have some property that s continuous in that it s preserved by taking the right kind of limit, you can show that an arbitrary function has this property by showing that the nicer ones do. Here are some examples. Measurable functions can be approximated pointwise by simple functions. If the limiting function is nonnegative (consider f + and f ), the simple functions can be chosen monotonically increasing, so the monotone convergence theorem can be used. Continuous functions on a compact interval can be approximated uniformly by step functions (linear combinations of characteristic functions of intervals), piecewise linear functions, or smooth functions. Also: polynomials, trigonometric polynomials, or anything else that satisfies the hypotheses of the Stone-Weierstrass theorem. You can also use rationals for most of these things, which is useful for showing something is separable. (E.g. if polynomials work, then so do polynomials with rational coefficients.) Convolution is a useful tool for approximating functions by smooth (C ) functions. Let φ be your favorite smooth function which integrates to 1. (I imagine something Gaussian-ish.) Let φ n (x) = nφ(nx), so that φ n gets a sharper and sharper bump at 0 while still integrating to 1. As a general principle, φ n f is a smooth function, and φ n f f in an appropriate sense. Specifically: If f is uniformly continuous then φ n f f uniformly. In particular this works if f is continuous with compact support. 1

2 If f C k c then φ n f f in C k c, i.e. the first k derivatives of φ n f converge uniformly to the corresponding derivatives of f. If f L p then φ n f f in the L p norm. If f is a distribution then φ n f f in the usual sense for distributions. In a function space (or indeed any space), look for dense subsets. In a Hilbert space, orthonormal bases are often useful, just as in R n ; any element is a limit of (finite!) linear combinations of the basis elements e n. If you prefer, you can rephrase any of these statements as something about a subset being dense. E.g. piecewise linear functions are dense in C([0, 1]), smooth functions are dense in L p. Deal with measurable sets in terms of generators. If a σ-algebra B is generated by a collection of sets E, and you are trying to prove something about all measurable sets in B, it is good to start by proving it holds for sets in E. But you should not then try to write an arbitrary measurable set in terms of sets from E by taking unions and complements; it can t be done in any simple way. 1 Instead, you can use the definition: if B = σ(e), then B is the smallest σ-algebra containing E, and so any other σ-algebra containing E will contain B as well. Thus, consider the collection of all sets that have the property you want. If you can show that it contains E and is (or contains) a σ-algebra, you ll be done. Pass to subsequences. If a sequence x n converges to x, then so do all of its subsequences x nk. Conversely, if every subsequence x nk has a further subsequence x nk j which converges to x, then x n itself also converges to x. (Intuitively, if x n did not converge to x, you could find a subsequence that converged to something else y, and all of its subsequences would converge to y as well.) This fact is commonly useful in connection with the theorem (2.30 in Folland) that if f n f in measure, there is a subsequence f nk such that f nk f a.e.; you can often take a theorem that requires a.e. convergence and weaken the hypothesis to convergence in measure (thus strengthening the theorem). Example 0.1. Show that the dominated convergence theorem holds for convergence in measure; i.e. if f n f in measure and f n g L 1 for all n, then f L 1 and f n f. 1 Technically, there is a way to do this, but you might have to take unions of complements of unions of complements, etc, and repeat the process up to an uncountable number of times. I don t recommend it unless you are very used to dealing with infinite ordinals and transfinite induction. 2

3 Proof. Let x n = f n, and let x nk be an arbitrary subsequence. The subsequence f nk also converges to f in measure (check this if you like). By the theorem above it has a subsequence of its own, f nk j which converges to f a.e. By dominated convergence f L 1 and f nk j f. So every subsequence x nk of x n has a further subsequence x nk j which converges to f, and therefore x n f as desired. Use the triangle inequality, in all its forms, generously. Any time you see an absolute value or norm on the outside of a sum, infinite sum, or integral, consider crashing through with it. More often than not it s the right thing to do, and if it s not, you can usually tell right away. Differences usually aren t as good an idea, since you are often subtracting two things that are close together, but consider it anyway. Another good trick for estimating differences is to add and subtract the same thing and then use the triangle inequality. Example 0.2. Let X, Y be normed spaces, x n x a convergent sequence in X, and T n T a norm-convergent sequence of bounded operators in L(X, Y). Show that T n x n T x in Y. Proof. Well, we d better show that T n x n T x Y 0. We have two things changing at once, so let s try to get them one at a time by adding and subtracting T x n. T n x n T x Y = T n x n T x n + T x n T x Y T n x n T x n Y + T x n T x Y T n T L(X,Y) x n X + T L(X,Y) x n x X Now the second term is going to zero because we know x n x. For the first term, note that since x n is a convergent sequence, it must be bounded (a very useful fact), so there is an M such that x n X M for all M. Since T n T L(X,Y) 0 by assumption, the first term is also going to zero, so we are done. Watch for the Big Three Baire theorems for Banach spaces. These are the Principle of Uniform Boundedness (PUB), the Open Mapping Theorem, and the Closed Graph Theorem. See section 5.3 of Folland. It can be easy to overlook an opportunity to use one of these theorems, which is unfortunate because they can be very powerful. I think it s because they are so very false in general normed spaces, and so they don t come so naturally to mind when thinking in normed-space terms. You may need to explicitly remind yourself when you have Banach spaces that these theorems are in play. 3

4 Shoot first and ask questions later. Real analysis has a lot of theorems of the form: Under [complicated set of conditions], [simple conclusion] holds. Perhaps the best examples are the integral convergence theorems and the Fubini/Tonelli theorems, which tell you when you can interchange various combinations of limits, integrals, and sums. When you want to make such an interchange, it s helpful to first try it and see if it gets you what you want. 2 If it does, then you can check the hypothesis of the theorem that justifies it. Work informally with distributions. Although the technical definitions for distributions are a bit complicated, we should try to remember that they are supposed to be generalized functions : they are designed to behave a lot like functions. So in the spirit of the previous item, it can be helpful in early stages of a problem to pretend that they are functions. Add them, differentiate them, integrate by parts (a biggie), convolve, take Fourier transforms. This will help you decide what the answer actually is, and you can then go back and verify it at a more technical level. Example 0.3. Compute the Fourier transforms of the Dirac delta distribution δ 0 and its derivative δ 0. Proof. Informally, the Fourier transform of δ 0 is supposed to be F [δ 0 ](ξ) = e ixξ δ 0 (x) dx. (1) δ 0 is supposed to be a point mass at 0, so this should be the value of e ixξ when x = 0; we expect F [δ 0 ](ξ) = 1. For δ 0, we integrate by parts : F [δ 0 ](ξ) = e ixξ δ 0 (x) dx ( d dx e ixξ ) δ 0 (x) dx = = iξe ixξ δ 0 (x) dx = iξ. Now that we know what the answers are, we ll check them more carefully. δ 0 is definitely a tempered distribution, so by definition F [δ 0 ] is the distribution 2 It almost always does. When I took 240 from Professor Bruce Driver, he propounded what we students came to call Driver s Theorem : Whenever you see two integrals, you should try to change their order. 4

5 such that F [δ 0 ], φ = δ 0, F [φ] for all φ S, the Schwartz class. But δ 0, F [φ] = F [φ](0) = e i0x φ(x) dx = φ(x) dx = 1, φ. (2) Next, F [δ 0 ], φ = δ 0, F [φ] = δ 0, F [φ] = F [φ] (0). But d dξ ξ=0 e ixξ φ(x) dx = = = d dξ e ixξ φ(x) dx ξ=0 ( ix)e ixξ φ(x) dx ξ=0 ixφ(x) dx = f, φ where f (x) = ix. The interchange of integral and derivative in the first line is an example of differentiating under the integral sign and should be justified; see Folland The theorem looks scary; it s perhaps easier just to remember the proof: it s just the definition of the derivative, the DCT and the mean value theorem. Well, perhaps that wasn t a great example; knowing the answer in advance didn t really help. But it did make for a good check. Maybe there is a better example. 5