Exercises

19¨C25 January

• 1.1: 1, 2, 3
• 1.2: 1¨C4, 8
• 1.4: 1¨C4, 6¨C8
• 1.6: 1¨C4
• 2.1: 1, 2, 4, 7
• 2.2: 1¨C5, 10, 9
• 2.3: 1, 2, 4, 9

26 January ¨C 1 February

• Prove Proposition 3.1.4 (the inverse triangle inequality): If \((X,d)\) is a metric space and \(x,y,z\in X\) then \(|d(x,y)-d(y,z)|\leq d(x,z)\).
• Section 3.1: Exercises 1, 2, 6, 7
• Section 3.2: Exercises 1, 2, 5, 6, 8
• Consider \(X=\mathbb{R}\) with the canonical metric \(d(x,y)=|x-y|\). Show that the interval?\((a,b)\) is open and that \([a,b]\) is closed,?for any \(a,b\in\mathbb{R}\) with \(a\leq b\). (What happens if \(a=b\)?) Explain why?\((a,b]\) is neither open nor closed when \(a<b\).

2¨C8 February

• Let \((X,d)\) be a metric space. Prove that every finite subset of \(X\) is closed. (A finite set is a set containing only finitely many points.)
  Hint:?Show first that singletons (i.e. sets containing only one point) are closed. Next, use Proposition 3.3.13 b).
• Section 3.3: Exercise 1, 2, 3, 11, 7
• Section 3.3: Let \((X,d_X)\) and \((Y,d_Y)\) be metric spaces, let \(f:X\to Y\) be a continuous function, let \(y\in Y\) be some given?point, and consider the problem of finding an \(x\in X\) such that \(f(x)=y\) (that is, we wish to solve the above equation). Prove that the set of solutions of this equation is a closed subset of \(X\).
  Hint:?Phrase the question in terms of finding \(f^{-1}\) of a closed set.
• Consider \(X=\mathbb{R}\) with the canonical metric \(d(x,y)=|x-y|\). The?support?of a function \(f:\mathbb{R}\to\mathbb{R}\) is the set of points \(x\) where \(f(x)\neq 0 \). Prove that the support of a continuous function is always open.
  Note: In the literature, the "support" of?\(f\)?is usually defined to be the closure of the above set, which is of course closed, not open.
• Section 3.4: Prove that \((\mathbb{Z},d)\), where \(d(x,y)=|x-y|\), is complete.
  Hint:?What does it mean for a sequence in \((\mathbb{Z},d)\) to converge or to be Cauchy?
• Section 3.4: Exercise 2, 4, 5, 6, 8
• Show that the equation?\(\cos t = 2t\)?has a unique solution.
  Hint: Formulate the problem as finding the fixed point of a function?\(f\).

9¨C15 February

• Section 3.5: Exercises 5¨C11 and 15.
• Section 4.1: Exercise 2, 4
• For each of the following concepts, write down its definition. If you know several definitions, write them all down.?Try to be as rigorous as you can: for example,?write "For every?\(\epsilon>0\)?there is some?\(\delta>0\)?such that¡­" and not "It's possible to fit a ball inside of¡­".?Repeat this exercise on at least three different days. For each time, don't open the book until you are finished.
  • A metric space.
  • A subset of a metric space is bounded.
  • An open subset of a metric space.
  • A closed subset of a metric space.
  • Convergence of a sequence in a metric space.
  • A function?f?from one metric space to another is continuous at a point?x.
  • A sequence in a metric space is Cauchy.
  • A complete metric space.

16¨C22 February

• Section 4.2: Exercises 1, 2, 5, 9
• For each of the following concepts, write down a definition. Try to be as rigorous as you can. Write your definitions both as rigorous definitions (as you would read in the book), and more informally using only words. Repeat this exercise at least three times on different days.
  • A subset of a metric space is not bounded.
  • A?subset of a metric space is not open.
  • A subset of a metric space is not closed.
  • A sequence in a metric space does not converge.
  • A function?f?from one metric space to another is discontinuous at a point?x.
  • A sequence in a metric space is not Cauchy.
  • A metric space is incomplete.
  • Hint:?The opposite of?\(\forall x : P(x)\)?("for all?x, the property?P(x)?is true") is?\(\exists x: \neg P(x)\)?("there exists an?x?such that?P(x)?is not true"). The opposite of?\(\exists x : P(x)\)?("there exists an?x for which?the property?P(x)?is true") is?\(\forall x: \neg P(x)\)?("for every?x, the property?P(x)?is not true").
• Section 4.5: Exercises 1, 2, 7
• Section 4.6: Exercises 1, 2, 3
• The space \(C_b(X,Y)\) is always "larger" than \(Y\), in the sense that \(Y\) can be embedded in \(C_b(X,Y)\):
  • Indeed, show that the map \(i:Y\to C_b(X,Y)\) which maps \(y\in Y\) to the constant function \(f(x)\equiv y\), is?an embedding (cf. Definition 3.1.3).
  • Show that \(i(Y)\) is precisely the subset of constant functions, and that this set is?a closed subset of \(C_b(X,Y)\).
  • Conclude that \(C_b(X,Y)\) is complete if and only if \(Y\) is complete.

23 February¨C1 March

• Section 4.3: Exercises 1 (note: this is a Fourier series!), 2, 3, 5
• Section 4.4: Exercises 1, 3, 4, 5
• Section 4.7: Exercises 1, 2

2¨C8 March

• Section 4.7: Let f be the function \(f(y,t)=y\). Choose initial data, say, \(\bar y=1\). Perform a fixed point iteration of the equation
  \(y(t) = \bar y + \int_0^t f(y(s),s)\,ds\)
  that is, for some continuous function \(y^0\)?(here, the superscripts are indices, not powers) let
  \(y^{n+1}(t) = \bar y + \int_0^t f(y^n(s),s)\,ds\)
  for \(n=0,1,2,\dots\). (It's easiest to start with \(y^0\equiv0\).) Compute \(y^1,y^2,y^3,y^4\). Give an expression for \(y^n\) for any \(n\), and prove that \(y^n\to y\), where \(y(t)=e^t\).
• Section 4.8: 3, 6, 7
• For each of the following concepts, write down its definition. If you know several definitions, write them all down.?Try to be as rigorous as you can: for example,?write "For every?\(\epsilon>0\)?there is some?\(\delta>0\)?such that¡­" and not "It's possible to fit a ball inside of¡­".?Repeat this exercise on at least three different days. For each time you do this, don't open the book until you are finished.
  • A metric space is compact
  • A subset of a metric space is compact
  • A set is dense in a metric space
  • A metric space is separable
  • A function from one metric space to another is bounded
  • A series of functions is uniformly convergent??????

9¨C15 March

• Section 4.10: 1, 2, 3
• Given?\(f\in C([0,1],\mathbb{R})\), the?n-th order Bernstein approximation of?f?is the polynomial
  \(f_n(x) = \sum_{k=0}^n {n \choose k} x^k (1-x)^{n-k} f(k/n).\)
  Using Matlab, Python or similar, compute and plot the Bernstein approximation of the following functions for various choices of?n:?\(f(x)=1,\ f(x)=x,\ f(x)=x^2,\ f(x)=|x-1/2|,\ f(x)=\sqrt{|x-1/2|}.\)?Prove that for the first two functions,?\(f=f_n\)for all \(n\).
• Prove Proposition 5.1.4.
• Exercise 5.1.10, 5.1.11.