mathematical_analysis
Introduction
數學分析
邏輯(Logic)
集合的運算
實數(Real number)
Dedekind分劃(cut)
Dedekind分劃2(cut)
集合(Set)
集合2(Set)
Cantor集合
Cauchy sequence
單調序列(monotonic sequence)
不等式(Ineqlity)
歐式空間(Euclidean space)
可數集合(Countable set)
歐式空間拓樸集(Euclidean space topology)
開集合(open set)
閉集合(close set)
稠密集合(dense set)
緊緻集合(compact set)
緊緻集合等價敘述(compact set equivment)
緊緻集合應用(compact set application)
連通集合(connected set
度量空間(Metric space)
Hausdorff空間
點拓撲集定義(Point topology definition)
點拓撲集定義2(Point topology definition 2)
點拓撲集理論(Point topology theorem)
序列(Sequence)
級數(Series)
函數(Function)
微分(Derivative)
微分2(Derivative)
向量微分(Derivative of vector)
有界變分(Bounded variation)
Riemann-Stieltjes可積分函數性質
Riemann-Stieltjes積分存在性
Riemann-Stieltjes積分存在性2
Riemann-Stieltjes積分微積分定理
Lebesgue積分
Sigma field(algebra)
測度(Measure)
可測函數(Measurable function)
隨機變數收斂性(Convergence of r.v.)
複數(Complex number)
向量空間(Vector space)
賦範空間(Normed space)
劣梯度(Subgradient)
Weierstrass theorems
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度量空間(Metric space)
度量空間 (Metric space)
Metric space
(
X
,
d
)
(X,d)
(
X
,
d
)
是由 set
X
X
X
與 distance function (metric)
d
d
d
兩者所組成。
d
:
X
×
X
→
[
0
,
∞
)
d: X \times X \rightarrow [0, \infty)
d
:
X
×
X
→
[
0
,
∞
)
,
∀
x
,
y
,
z
∈
X
\forall x,y,z \in X
∀
x
,
y
,
z
∈
X
satisfies
(Non-negativity, 非負性)
d
(
x
,
y
)
=
0
⇔
x
=
y
d(x,y)=0 \Leftrightarrow x=y
d
(
x
,
y
)
=
0
⇔
x
=
y
.
(Symmetry, 距離有對稱性)
d
(
x
,
y
)
=
d
(
y
,
x
)
d(x,y)=d(y,x)
d
(
x
,
y
)
=
d
(
y
,
x
)
(Triangle inequality, 三角不等式)
d
(
x
,
y
)
+
d
(
y
,
z
)
≤
d
(
x
,
z
)
d(x,y)+d(y,z) \leq d(x,z)
d
(
x
,
y
)
+
d
(
y
,
z
)
≤
d
(
x
,
z
)
.
度量與賦範的比較(Comparison between metric and norm)
Metric
Norm
binary operator
d
:
X
×
X
→
R
+
d: X \times X \rightarrow \mathbb{R}^{+}
d
:
X
×
X
→
R
+
uniary operator:
∥
⋅
∥
:
X
→
R
+
\begin{Vmatrix} \cdot \end{Vmatrix}: X \rightarrow \mathbb{R}^{+}
∥
∥
⋅
∥
∥
:
X
→
R
+
d
(
x
,
y
)
=
0
⇔
x
=
y
d(x,y) = 0 \Leftrightarrow x=y
d
(
x
,
y
)
=
0
⇔
x
=
y
∥
x
∥
=
0
⇔
x
=
0
\begin{Vmatrix} x \end{Vmatrix} =0 \Leftrightarrow x=0
∥
∥
x
∥
∥
=
0
⇔
x
=
0
d
(
x
,
y
)
=
d
(
y
,
x
)
d(x,y) = d(y,x)
d
(
x
,
y
)
=
d
(
y
,
x
)
∥
c
x
∥
=
∣
c
∣
∥
x
∥
\begin{Vmatrix} cx\end{Vmatrix} =\begin{vmatrix}c\end{vmatrix}\begin{Vmatrix} x \end{Vmatrix}
∥
∥
c
x
∥
∥
=
∣
∣
c
∣
∣
∥
∥
x
∥
∥
d
(
x
,
y
)
+
d
(
y
,
z
)
≤
d
(
x
,
z
)
d(x,y)+ d(y,z) \leq d(x,z)
d
(
x
,
y
)
+
d
(
y
,
z
)
≤
d
(
x
,
z
)
∥
x
+
y
∥
≤
∥
x
∥
+
∥
y
∥
\begin{Vmatrix} x+ y \end{Vmatrix} \leq \begin{Vmatrix} x \end{Vmatrix} + \begin{Vmatrix} y \end{Vmatrix}
∥
∥
x
+
y
∥
∥
≤
∥
∥
x
∥
∥
+
∥
∥
y
∥
∥
.
常見的metric space
(
Euclidean space
)
d
(
x
,
y
)
=
∣
x
−
y
∣
=
∑
n
=
1
N
(
x
n
−
y
n
)
2
d(x,y)=|x-y| = \sqrt{\sum_{n=1}^N (x_n - y_n)^2 }
d
(
x
,
y
)
=
∣
x
−
y
∣
=
√
∑
n
=
1
N
(
x
n
−
y
n
)
2
on
R
n
\mathbb{R}^n
R
n
.
(
Discrete space
)
ϕ
⊂
M
\phi \subset M
ϕ
⊂
M
,
d
(
x
,
y
)
=
1
d(x,y)=1
d
(
x
,
y
)
=
1
if
x
=
y
x=y
x
=
y
else
d
(
x
,
y
)
=
0
d(x,y)=0
d
(
x
,
y
)
=
0
.
(
Functional space
)
f
,
g
f, g
f
,
g
are continuous functions
d
1
(
f
,
g
)
=
∫
0
1
∣
f
(
x
)
−
g
(
x
)
∣
d
x
d_{1}(f,g)=\int_{0}^{1}\left|f(x)-g(x)\right|dx
d
1
(
f
,
g
)
=
∫
0
1
∣
f
(
x
)
−
g
(
x
)
∣
d
x
d
∞
(
f
,
g
)
=
sup
0
≤
x
≤
1
∣
f
(
x
)
−
g
(
x
)
∣
d_{\infty}(f,g)=\sup_{0\leq x\leq1}\left|f(x)-g(x)\right|
d
∞
(
f
,
g
)
=
sup
0
≤
x
≤
1
∣
f
(
x
)
−
g
(
x
)
∣
完備度量空間 (Complete metric space)
A sequence
{
x
n
}
\left\{ x_{n}\right\}
{
x
n
}
in
(
X
,
d
)
\left(X,d\right)
(
X
,
d
)
is called
Cauchy
if
d
(
x
n
,
x
m
)
→
0
d\left(x_{n},x_{m}\right)\rightarrow0
d
(
x
n
,
x
m
)
→
0
as
n
,
m
→
∞
n,m\rightarrow\infty
n
,
m
→
∞
.
A converge sequence is a Cauchy sequence. (收斂數列必為Cauchy數列,反之不一定成立,因為極限點不一定在
X
X
X
中)
A Cauchy sequence in the complete space is a converge sequence
.
Complete metric space是space中任意的Cauchy數列都會收斂, 且數列的極限值在
X
X
X
中。
E.g. Rational number
Q
\mathbf{Q}
Q
with
d
(
x
,
y
)
=
∣
x
−
y
∣
d(x,y)=|x-y|
d
(
x
,
y
)
=
∣
x
−
y
∣
不是complete metric space,因為
{
1
.
4
,
1
.
4
1
,
1
.
4
1
4
,
1
.
4
1
4
2
,
⋯
}
\left\{ 1.4,1.41,1.414,1.4142,\cdots\right\}
{
1
.
4
,
1
.
4
1
,
1
.
4
1
4
,
1
.
4
1
4
2
,
⋯
}
此Cauchy數列收斂,但極限點並非有理數。
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