math_review

topics

mathmatics

1. supreme and optimum
2. Norm and Linear product
3. topology of R^*
4. Continuious Function

supreme and optimum

Def 1: 非空有界集合必有上确界

common norm

(1) x$\in$ Rⁿ, ||x||₂= $\sqrt{x_1^2+x_2^2+...+x_n^2}$
(2) x$\in$ R

Norm of vector

Def 2: a function f : Rⁿ $\to$ R is called norm,if
(1) f(x)>=0 & f(x)=0 if x=$\vec{0}$
(2) homogenous: f(tx)=|t| f(x)
(3) triangle inquality: f(x+y)<=f(x)+f(y)
f(x)=||x||_k

common norm

(1) x$\in$ Rⁿ, ||x||₂= $\sqrt{x_1^2+x_2^2+...+x_n^2}$
(2) x$\in R^n$, $||x|{|}_p=(x_1^p+...+x_n^p{)}^{\frac{1}{p}}$
(2) xKaTeX parse error: Got function '\inf' with no arguments as subscript at position 15: \in R^n,||x||_\̲i̲n̲f̲=max\{|x_1|,...…

Def 3: Unit Ball
$B=x\in R^n,||x||\le 1$

norm of matrix

$P\in S_{++}^n,||x|{|}_p=(<x,P_x>{)}^{\frac{1}{2}}$

1. KaTeX parse error: Got function '\sum' with no arguments as argument to '\sqrt' at position 15: ||A||_F=\sqrt \̲s̲u̲m̲_{i=1}^m\sum_{j…
2. $||A|{|}_p=\underset{||x|{|}_p=1}{\max }||Ax|{|}_p$
3. $||A||=\underset{j}{\max }{\sum }_i|aij|$
4. $||A|{|}_2=\sigma (A)$(奇异值,特征值绝对值最大)
5. KaTeX parse error: Got function '\inf' with no arguments as subscript at position 7: ||A||_\̲i̲n̲f̲=\underset i{ma…

Inner product

f：$R^n\odot R^n\to R$

1. Nonnegative: $f(x,x)\ge 0,f(x,x)=0iffx=0$
2. Symmetric:$f(x,y)=f(y,x)$
3. Bilinear:$f(ax+by,z)=af(x,z)+bf(y,z)$
   $f(z,ax+by)=af(z,x)+bf(z,y)$
   f(x,y)=<x,y>

定义了内积of vector，we can find corresponding norm;法
$<x,x>=||x|{|}_2^2$，$<x,x{>}_p=||x|{|}_p^2$
given norm ,we can’t find corresponding inner product always(like KaTeX parse error: Got function '\inf' with no arguments as subscript at position 7: ||x||_\̲i̲n̲f̲)

example of inner product

• comply with above 3 properties

1. $A,B\in R^{m+n},<A,B>=tr(A^{T}B)={\sum }_{ij}{a}_{ij}{b}_{ij}$
2. $l^2(R)=x=(x_1,...),x_1\in R,{\sum }_{i=1}^n|x{|}^2<\inf$
   KaTeX parse error: Got function '\inf' with no arguments as superscript at position 18: …,y>=\sum_{i=1}^\̲i̲n̲f̲ ̲x_iy_i < \inf
3. $L^2(R)=f:R\to R,{\int }_R|f(x){|}^{2}dx<\inf$
   $<f,g{>}_{L^2}={\int }_{R}f(x)g(x)dx$

Def 5: $x,y\in R^n$
included angle，$ang(x,y)=arccos\frac{<x,y>}{||x|||.||y||}$

Topology of IRⁿ

logistic chain:$Norm\to UnitBall\to Neighbourhood\to Limit$

Def 6: given $ϵ$,the $ϵ$-neighbourhood of a point x
$N_{ϵ}(x)=y,y\in I{R}^n,||y-x||<ϵ$
Def 7: interor point of S,$\exists ϵ>0,N_{ϵ}(x)\in S$
Def 8: interior of S,the set of all interior points of S
Def 9: Open set O=int O
Def 10: Closed Set IR\F is open
Def 11: $(x_k)$ converges to x,if $\forall ϵ>0,\forall p,||x_k-x|{|}_p<ϵ,ifk>N$
Def 12:(limited point) $x\in I{R}^n,S\in I{R}^n,\exists (x_k),x_k\ne x,(x_k)\in S,s.t.x_k\to x$
Def 13: F is closed if it c ontains all its limit points