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 $x$.}