MATH 522 (Analysis
II)
Definitions and Theorems
Ruixuan
Tu
ruixuan.tu@wisc.edu
University of Wisconsin-Madison
Joris Ross & Andreas Seeger, 25 January 2023
Walter Rudin: Principles of Mathematical Analysis
[Thm] A subset \(A\) of \(\mathscr{c}_0\) is totally bounded if the sequence \(n\to \sup_{a\in A} a_n\) converges to zero
[Lemma] Suppose \(A\subset X\) and assume that for every \(\varepsilon > 0\), there is a finite number of balls \(B(z, \varepsilon)\), \(z\in Z\) (\(Z\): finite set in \(\mathscr{c}_0\)), \(\# z<\infty\). \(A\subset \cup_{z\in Z} B(z, \varepsilon)\), then \(A\) is totally bounded
[Prop] Given \(\varepsilon>0\), find a finite collection of balls \(B(z, \frac{\varepsilon}{2})\), \(z\in Z\) (finite). Let \(Z'\subset Z\) be the collection of those \(z\in Z\) for which \(B(z, \frac{\varepsilon}{2})\cap A\)
For \(z\in Z'\) there is an \(a_z\in A\cap B(z, \frac{\varepsilon}{2})\), \(B(a_z, \varepsilon)\supset B(a_z, \frac{\varepsilon}{2})\)
If \(y\in B(z, \frac{\varepsilon}{2}), d(y, z)<\frac{\varepsilon}{2}, d(a_z, z)<\frac{\varepsilon}{2}\), then by triangle inequality \(d(y, a_z)<\varepsilon\)
[Def] pointwise bounded functions: A class \(A\) of functions is pointwise bounded if for every \(x\) there is \(M_x<\infty\) such that \(\sup_{f\in A} \left\vert f(x) \right\vert \leq M_x\)
[Def] uniformly bounded functions: \(A\) is uniformly bounded if \(\sup_{x\in X} \sup_{f\in A} \left\vert f(x) \right\vert \leq M_C\)
[Rmk] If \(A\) is uniformly bounded, \(A\) may not be totally bounded
[Fact] totally bounded \(\implies\) uniformly bounded (Linrong)
[Def] equicontinuous: \(A\subset C(K)\) is equicontinuous if for every \(\varepsilon\) there exists a \(\delta\) such that \(\left\vert f(x)-f(y) \right\vert < \varepsilon\) for all \(f\in A\) and for all \(x, y\) such \(d(x, y)<\delta\)
[Thm] Arzelà–Ascoli: \(A\subset C(K)\) is totally bounded if and only if \(A\) is pointwise bounded and equicontinuous
Proof: \(w_{\delta}(f)=\sup_{d(x, y)< \delta} d(f(x), f(y))\), \(\sup_{f\in A} w_{\delta}(f)\to 0\) as \(\delta \to 0\)
[Rmk] Arzelà–Ascoli theorem characterizes the totally bounded sets \(f: K\to \mathbb{R} \text{ or } \mathbb{C}\) in the sense \(C(K)\) (continuous functions on a compact set \(K\)), \(\left\| f \right\|=\sup_{x\in K} \left\vert f(x) \right\vert\)
[Rmk] If \(A\subset C(K)\) is equicontinuous, then \(A\) is pointwise bounded if and only if it is uniformly bounded
[Def] contraction: If \((X, d)\) is a metric space then \(T: X\to X\) is called a contraction if there exists \(\alpha<1\) such that \(d(T(x), T(y))\leq \alpha d(x, y)\)
[Thm] Banach fixed point theorem (aka contraction principle): If \((X, d)\) is complete, then any contraction have a unique fixed point, i.e., a unique \(x\in X\) such that \(T(x)=x\)
[Proof] method of successive approximation (construct a sequence, pick \(x_0\), then \(x_1=T(x_0), x_2=T(x_1), \dots, x_{n+1}=T(x_n)\)); \(d(x_{n+1}, x_n)=d(T(x_n), T(x_{n-1}))\leq \alpha d(x_n, x_{n-1})\leq \alpha \cdot \alpha^{n-1} d(x_1, x_0)\) \(\implies\) \(d(x_n, x_{n-1})\leq \alpha^{n-1} d(x_0, x_1)\)
[Def] initial value problem for differential equation
Pick a point \((x_0, y_0)\) to be initial point,
For a function \((x, y)\mapsto F(x, y)\) continuous near \(x_0, y_0\), we define the initial value problem
[Thm] Peano existence theorem: If \(F\) is continuous near \((x_0, y_0)\), then the initial value problem has a solution (based on compactness)
[Thm] Picard–Lindelöf theorem: If in addition there is a condition \(\left\vert F(x, y)-F(x, \tilde{y}) \right\vert \leq C\left\vert y-\tilde{y} \right\vert\) for \((x, y)\) and \((x, \tilde{y})\) near \((x_0, y_0)\); then the solution is unique (based on the contraction principle)
[Thm] Picard-Lindelöf theorem: Let \(D\subseteq \mathbb{R}\times \mathbb{R}^n\) be a closed rectangle with \((t_0, y_0)\in D\). Let \(f: D\to \mathbb{R}^n\) be a function that is continuous in \(t\) and Lipschitz continuous in \(y\). Then there exists some \(c>0\) such that the initial value problem has a unique solution \(y(t)\) on the interval \(\left[ x_0-\delta, x_0+\delta \right]\) (from Wikipedia)
[Def] Lipschitz continuity: Given two metric spaces \((X, d_X)\) and \((Y, d_Y)\), a function \(f: X\to Y\) is called Lipschitz continuous if there exists a real constant \(K\geq 0\) such that, for all \(x_1\) and \(x_2\) in \(X\), \(d_Y(f(x_1), f(x_2))\leq Kd_X(x_1, x_2)\). Any such \(K\) is a Lipschitz constant. If \(0\leq K<1\) and \(f\) maps a metric space to itself, the function is called a contraction (from Wikipedia)
[Def] Lipschitz condition: Function \(f(t, y)\) satisfies a Lipschitz condition in the variable \(y\) on a set \(D\subseteq \mathbb{R}^2\) if a constant \(L>0\) exists with \(\left\vert f(t, y_1)-f(t, y_2) \leq L \left\vert y_1-y_2 \right\vert \right\vert\) whenever \((t, y_1)\), \((t, y_2)\) are in \(D\). \(L\) is Lipschitz constant (from UCB Math)
[Lemma] For a continuous function \(x\to y(x)\) defined on \(\left[ x_0-\delta, x_0+\delta \right]\) with values in \(\left[ y_0-b, y_0+b \right]\). The following two conditions are equivalent: