Homework 1

Deadline: Oct. 7, 22:00:00

Norms, linear space and convergence

1. We mentioned in the last lecture that in $\mathbb{R}^n$ if $\{x_n\}$ converge to $x$ with one norm, then ${x_n}$ converge to $x$ with any norm.

   We now consider a special case, $\ell_p$-norms.

   (1). Show that for any $p \geq 1$ and any $x\in \mathbb{R}^n$, $\lVert x \rVert_\infty \leq \lVert x\rVert_p \leq n\cdot \lVert x \rVert_\infty$.

   (2). Show that for any $p, q\geq 1$, if $\{x_n\}\to x$ with norm $\lVert \cdot \rVert_p$ then $\{x_n\}\to x$ with norm $\lVert \cdot \rVert_q$.

Compact sets and extremum value

2. Does $f(x_1,x_2,x_3)=3x_1^3+x_1x_2+e^{−x_3}+\ln⁡|x_1+x_2x_3|$ have global maximum or minimum value on the set $\{(x_1,x_2,x_3):(x_1-2x_2)^2+(3x_1+x_3)^2≤2022, x_1+x_2x_3\neq 0\}$ ?

Differential and gradients

3. Compute the differentials of functions $g(r,θ,φ)=(r\sin ⁡θ \cos ⁡φ, r\sin ⁡θ \sin ⁡φ,r\cos ⁡θ)^\mathsf{T}$ and $h(x,y,z)=(\sqrt{x^2+y^2+z^2},\arccos ⁡\frac{z}{\sqrt{x^2+y^2+z^2}},\arctan ⁡\frac{y}{x})^\mathsf{T}$.

   What is the relation between $g$ and $h$, and what is the relation between $Dh$ and $Dg$ ?