Isaiah Bascom

Exercises from Real Analysis with worked solutions.

Exercise 1. Question 4.1.A

§4.1HW 1

\text{Establish the Pythagorean formula: if $x$ and $y$ are} \\ \text{orthogonal vectors,} \\ \text{prove that $\lVert x+y\rVert = \left(\lVert x\rVert^2+\lVert y\rVert^2\right)^{1/2}$.}

Exercise 2. Question 4.1.C

§4.1HW 1

\text{Show that $\lVert x+y\rVert^2+\lVert x-y\rVert^2 = 2\lVert x\rVert^2+2\lVert y\rVert^2$ for} \\ \text{all vectors $x,y\in\mathbb{R}^n$.} \\ \text{Geometric meaning?}

Exercise 3. Question 4.1.D

§4.1HW 1

\text{Prove that if $x,y\in\mathbb{R}^n$ then $\big|\lVert x\rVert-\lVert y\rVert\big|\le\lVert x-y\rVert$.}

Exercise 4. Question 4.1.E

§4.1HW 1

\text{Prove by induction that} \\ \text{$\lVert x_1+\cdots+x_k\rVert\le\lVert x_1\rVert+\cdots+\lVert x_k\rVert$ for vectors} \\ \text{$x_i\in\mathbb{R}^n$.}

Exercise 5. Question 4.1.F

§4.1HW 1

\text{Suppose that $x$ and $y$ are unit vectors in $\mathbb{R}^n$. Show} \\ \text{that if $\big\lVert\frac{x+y}{2}\big\rVert=1$, then $x=y$.}

Exercise 6. Question 4.1.I

§4.1HW 1

\text{Suppose that $U$ is a linear transformation from $\mathbb{R}^n$ to} \\ \text{$\mathbb{R}^m$ that is \textit{isometric},} \\ \text{meaning that $\lVert Ux\rVert=\lVert x\rVert$ for all $x\in\mathbb{R}^n$.} \\[0.5em] \text{(a) Prove that $\langle Ux,Uy\rangle=\langle x,y\rangle$ for all $x,y\in\mathbb{R}^n$.} \\ \text{(b) If $\{v_1,\dots,v_n\}$ is an orthonormal set in $\mathbb{R}^m$, show} \\ \text{that the linear transformation} \\ \text{$Ux=\sum_{i=1}^n x_iv_i$ is isometric.}

Exercise 7. Question 4.1.K

§4.1HW 1

\text{Let $M$ be a subspace of $\mathbb{R}^n$ with an orthonormal basis} \\ \text{$\{v_1,\dots,v_k\}$.} \\ \text{Define a linear transformation on $\mathbb{R}^n$ by} \\ Px=\sum_{i=1}^k\langle x,v_i\rangle v_i. \\[0.5em] \text{(a) Show that $Px$ belongs to $M$, and $Py=y$ for all} \\ \text{$y\in M$. Hence show that $P^2=P$.} \\ \text{(b) Show that $\langle Px,x-Px\rangle=0$.} \\ \text{(c) Hence show that $\lVert x\rVert^2=\lVert Px\rVert^2+\lVert x-Px\rVert^2$.} \\ \text{(d) If $y\in M$, show that} \\ \text{$\lVert x-y\rVert^2=\lVert y-Px\rVert^2+\lVert x-Px\rVert^2$.} \\ \text{(e) Hence show that $Px$ is the closest point in $M$ to $x$.}

Exercise 8. Question 4.2.A

§4.2HW 2

\text{(a) If $(x_n)^{\infty}_{n=1}$ is a sequence in $\mathbb{R}^n$ with $\lim_{n \to \infty} \mathbf{x_n} = \mathbf{a}$,} \\ \text{show that $\lim_{n \to \infty} \lVert \mathbf{x}_n \rVert = \lVert \mathbf{a} \rVert$.} \\[0.5em] \text{(b) Show by example that the converse is false.}

Exercise 9. Question 4.2.B

§4.2HW 2

\text{Show that if $(\mathbf{x}_k)$ is a sequence in $\mathbb{R}^n$ such that} \\ \text{$\sum_{k \geq 1} \lVert \mathbf{x}_k - \mathbf{x}_{k+1} \rVert < \infty$,} \\ \text{then $(\mathbf{x}_k)$ is a Cauchy sequence.}

Exercise 10. Question 4.2.D

§4.2HW 2

\text{Let $x_0 \in \mathbb{R}^n$ and $R > 0$. Prove that} \\ \text{$\{x \in \mathbb{R}^n : \|\mathbf{x} - \mathbf{x_0}\| \leq R \}$ is complete.}

Exercise 11. Question 4.2.E

§4.2HW 2

\text{Let $M$ be a subspace of $\mathbb{R}^n$.} \\[0.5em] \text{(a) Let $\{\mathbf{v}_1, \dots, \mathbf{v}_m\}$ be an orthonormal basis for $M$.} \\ \text{Formulate an analogue of Lemma 4.2.3 for $M$ and} \\ \text{prove it.} \\[0.5em] \text{(b) Prove that $M$ is complete.}

Exercise 12. Question 4.2.F

§4.2HW 2

\text{Let $\mathbf{v}_0 = (x_0, y_0)$ with $0 < x_0 < y_0$. Define} \\ \mathbf{v}_{n+1} = (x_{n+1}, y_{n+1}) = \left( \sqrt{x_n y_n}, \ \frac{x_n + y_n}{2} \right) \\ \text{for all $n \geq 0$.} \\[0.5em] \text{(a) Show by induction that} \\ \text{$0 < x_n < x_{n+1} < y_{n+1} < y_n$.} \\ \text{(b) Then estimate $y_{n+1} - x_{n+1}$ in terms of $y_n - x_n$.} \\ \text{(c) Thereby show that there is a number $c$ such that} \\ \text{$\lim_{n \to \infty} \mathbf{v}_n = (c,c)$.}

Exercise 13. Question 4.2.H

§4.2HW 2

\text{Let } T = \begin{bmatrix} 5/4 & -1/4 \\ 3/4 & 1/4 \end{bmatrix}\text{. Set $\mathbf{x}_n = T^n(1,0)$ for $n \geq 1$.} \\[0.5em] \text{(a) Prove that $(\mathbf{x}_n)$ converges and find the limit $\mathbf{y}$.} \\ \text{(b) Find an explicit $N$ so that $\|\mathbf{x}_n - \mathbf{y}\| < \frac{1}{2} 10^{-100}$ for} \\ \text{all $n \geq N$.}

Exercise 15. Example of Convergence in ℝ²

§4.2

\text{Let } \mathbf{v}_0 = \begin{pmatrix} 0 \\ 0 \end{pmatrix} \text{ and define a sequence} \\ \mathbf{v}_n = \begin{pmatrix} x_n \\ y_n \end{pmatrix} \in \mathbb{R}^2 \text{ recursively by} \\ x_{n+1} = \frac{x_n + y_n + 1}{2}, \quad y_{n+1} = \frac{x_n - y_n + 1}{2}. \\[0.5em] \text{Does } (\mathbf{v}_n) \text{ converge?}