Homework 1

Due: Sept. 30, 14:00:00

Norms, linear space and convergence

We would like to show that in a finite dimensional linear space, if $\{x_n\}$$\{x_n\}$ converge to $x$$x$ with respect to one norm, then $\{x_n\}$$\{x_n\}$ converge to $x$$x$ with respect to any norm.

Suppose that $V$$V$ is a finite dimensional linear space (you can assume $V={\mathbb{R}}^n$$V = \mathbb{R}^n$), $\Vert \cdot {\Vert }_a$$\lVert \cdot \rVert_a$ and $\Vert \cdot {\Vert }_b$$\lVert \cdot \rVert_b$ are two norms.

1. Show that $\Vert \cdot {\Vert }_b:V\to \mathbb{R}$$\lVert \cdot \rVert_b : V\to \mathbb{R}$ is a bounded function on the set $\{v:\Vert v{\Vert }_a=1\}$$\{v:\lVert v\rVert_a=1\}$.
2. Show that $\Vert \cdot {\Vert }_b:V\to \mathbb{R}$$\lVert \cdot \rVert_b : V\to \mathbb{R}$ is a continuous function with respect to norm $\Vert \cdot {\Vert }_a$$\lVert \cdot \rVert_a$. Namely, for any $v\in V$$v\in V$, and any $\epsilon >0$$\varepsilon > 0$, there exists $\delta >0$$\delta >0$ such that for all $u\in V$$u\in V$, if $\Vert v-u{\Vert }_a<\delta$$\lVert v-u\rVert_a<\delta$ then $\Vert v-u{\Vert }_b<\epsilon$$\lVert v-u\rVert_b<\varepsilon$.

Then it is a simple corollary that $\{x_n\}\to x$$\{x_n\} \to x$ in norm $\Vert \cdot {\Vert }_a$$\lVert \cdot \rVert_a$ implies $\{x_n\}\to x$$\{x_n\} \to x$ in norm $\Vert \cdot {\Vert }_b$$\lVert \cdot \rVert_b$.

Compact sets and extremum value

3. Does $f(x_1,x_2,x_3)=3x_1^3+x_1{x}_2+e^{-x_3}+\ln |x_1+x_2{x}_3|$$f(x_1,x_2,x_3) = 3x_1^3+x_1x_2+e^{-x_3}+\ln \abs{x_1+x_2x_3}$ have extreme value on the set $\{(x_1,x_2,x_3):x_1^2+2x_1{x}_2+2x_2^2-x_1{x}_3-2x_2{x}_3+x_3^2\le 10\}$$\{(x_1,x_2,x_3):x_1^2 + 2 x_1 x_2 + 2 x_2^2 - x_1x_3 - 2 x_2x_3 + x_3^2 \leq 10\}$ ?

Differential and gradients

4. Compute the differentials of functions $g(r,\theta ,\phi )=(r\sin \theta \cos \phi ,r\sin \theta \sin \phi ,r\cos \theta {)}^{\mathrm{T}}$$g(r, \theta, \varphi) = (r \sin\theta\cos\varphi, r\sin\theta\sin\varphi,r\cos\theta)^{\mathrm{T}}$ and $h(x,y,z)=(\sqrt{x^2+y^2+z^2},\arccos \frac{z}{\sqrt{x^2+y^2+z^2}},\arctan \frac{y}{x}{)}^{\mathrm{T}}$$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})^{\mathrm{T}}$.

   What is the relationship between $Dh$$Dh$ and $Dg$$Dg$ ?