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news 2025/7/18 22:28:39

topics

mathmatics

supreme and optimum
Norm and Linear product
topology of R^*
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 $\rightarrow$ 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) x $KaTeX parse error: Got function '\inf' with no arguments as subscript at position 15: \in R^n,||x||_\̲i̲n̲f̲=max\{|x_1|,...…$

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

norm of matrix

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

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

Inner product

f： $R^n \odot R^n \rightarrow R$

Nonnegative: $f(x,x)\geq 0,f(x,x)=0 iff x=0$
Symmetric: $f (x, y) = f (y, x)$
Bilinear: $f (a x + b y, z) = a f (x, z) + b f (y, z)$
$f (z, a x + b y) = a f (z, x) + b f (z, y)$
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

$A,B\in R^{m+n},<A,B>=tr(A^TB)=\sum_{ij}a_{ij} b_{ij}$
$l^2(R)={x=(x_1,...),x_1\in R,\sum_{i=1}^n |x|^2<\inf}$
$KaTeX parse error: Got function '\inf' with no arguments as superscript at position 18: …,y>=\sum_{i=1}^\̲i̲n̲f̲ ̲x_iy_i < \inf$
$L^2(R)={f:R\rightarrow R,\int_R|f(x)|^2 dx<\inf}$
$<f,g>_{L^2}=\int_Rf(x)g(x)dx$

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

Topology of IRⁿ

logistic chain: $Norm\rightarrow Unit Ball\rightarrow Neighbourhood\rightarrow Limit$

Def 6: given $\epsilon$ ,the $\epsilon$ -neighbourhood of a point x
$N_{\epsilon}(x)={y,y\in IR^n,||y-x||<\epsilon}$
Def 7: interor point of S, $\exists \epsilon>0,N_{\epsilon}(x)\in S$
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 $\forall \epsilon>0,\forall p,||x_k-x||_p<\epsilon,if\ k>N$
Def 12:(limited point) $x\in IR^n,S\in IR^n,\exists (x_k),x_k\neq x,(x_k)\in S,s.t. x_k\rightarrow x$
Def 13: F is closed if it c ontains all its limit points