数学 - Python - 解析学 - 連続写像の空間 - ノルム空間(定義、n次元実数空間、色々なノルム)

解析入門〈3〉

学習環境

• Surface 3 (4G LTE)、Surface 3 タイプ カバー、Surface ペン(端末)
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• 参考書籍
  • Pythonからはじめる数学入門
  • JavaScriptリファレンス 第6版
  • インタラクティブ・データビジュアライゼーション

解析入門〈3〉(松坂 和夫(著)、岩波書店)の第13章(連続写像の空間)、13.1(ノルム空間)、問題1.を取り組んでみる。

1. • 1. 
        
        ${\left|x\right|}_1=\underset{i=1}{\overset{n}{\sum \; <!--\; n-ary\; summation\; -->}}\left|x_i\right|\ge \; <!--\; greater-than\; or\; equal\; to\; -->0$
     2. 
        
        ${\left|x\right|}_1=\underset{i=1}{\overset{n}{\sum \; <!--\; n-ary\; summation\; -->}}\left|x_i\right|=0\Rightarrow \; <!--\; rightwards\; double\; arrow\; -->\left|x_1\right|=\dots \; <!--\; horizontal\; ellipsis\; -->=\left|x_n\right|=0$

        よって、

        $x_1=\dots \; <!--\; horizontal\; ellipsis\; -->=x_n=0$

        となるので、

        $x=0$
     3. 
        
        ${\left|cx\right|}_1=\underset{i=1}{\overset{n}{\sum \; <!--\; n-ary\; summation\; -->}}\left|c{x}_i\right|=\left|c\right|\underset{i=1}{\overset{n}{\sum \; <!--\; n-ary\; summation\; -->}}\left|x_i\right|=\left|c\right|{\left|x\right|}_1$
     4. 
        
        $\begin{array}{}{\left|x+y\right|}_1\\ =\underset{i=1}{\overset{n}{\sum \; <!--\; n-ary\; summation\; -->}}\left|x_i+y_i\right|\\ \le \; <!--\; less-than\; or\; equal\; to\; -->\underset{i=1}{\overset{n}{\sum \; <!--\; n-ary\; summation\; -->}}\left(\left|x_i\right|+\left|y_i\right|\right)\\ =\underset{i=1}{\overset{n}{\sum \; <!--\; n-ary\; summation\; -->}}\left|x_i\right|+\underset{i=1}{\overset{n}{\sum \; <!--\; n-ary\; summation\; -->}}\left|y_i\right|\\ ={\left|x\right|}_1+{\left|y\right|}_1\end{array}$
   • 1. 
        
        ${\left|x\right|}_{\mathrm{\infty \; <!--\; infinity\; -->}}=\max \left\{\left|x_1\right|,\dots \; <!--\; horizontal\; ellipsis\; -->,\left|x_n\right|\right\}\ge \; <!--\; greater-than\; or\; equal\; to\; -->0$
     2. 
        
        ${\left|x\right|}_{\mathrm{\infty \; <!--\; infinity\; -->}}=\max \left\{\left|x_1\right|,\dots \; <!--\; horizontal\; ellipsis\; -->,\left|x_n\right|\right\}=0\Rightarrow \; <!--\; rightwards\; double\; arrow\; -->\left|x_1\right|=\dots \; <!--\; horizontal\; ellipsis\; -->=\left|x_n\right|=0$

        よって、

        $x_1=\dots \; <!--\; horizontal\; ellipsis\; -->=x_n=0$

        となるので、

        $x=0$
     3. 
        
        $\begin{array}{}{\left|cx\right|}_{\mathrm{\infty \; <!--\; infinity\; -->}}=\max \left\{\left|c{x}_1\right|,\dots \; <!--\; horizontal\; ellipsis\; -->,\left|c{x}_n\right|\right\}\\ =\max \left\{\left|c\right|\left|x_1\right|,\dots \; <!--\; horizontal\; ellipsis\; -->,\left|c\right|\left|x_n\right|\right\}\\ =\left|c\right|\max \left\{\left|x_1\right|,\dots \; <!--\; horizontal\; ellipsis\; -->,\left|x_n\right|\right\}\\ =\left|c\right|{\left|x\right|}_{\mathrm{\infty \; <!--\; infinity\; -->}}\end{array}$
     4. 
        
        $\begin{array}{}{\left|x+y\right|}_{\mathrm{\infty \; <!--\; infinity\; -->}}\\ =\max \left\{\left|x_1+y_1\right|,\dots \; <!--\; horizontal\; ellipsis\; -->,\left|x_n+y_n\right|\right\}\\ \le \; <!--\; less-than\; or\; equal\; to\; -->\max \left\{\left|x_1\right|+\left|y_1\right|,\dots \; <!--\; horizontal\; ellipsis\; -->,\left|x_n\right|+\left|y_n\right|\right\}\\ \le \; <!--\; less-than\; or\; equal\; to\; -->\max \left\{\left|x_1\right|,\dots \; <!--\; horizontal\; ellipsis\; -->,\left|x_n\right|\right\}+\max \left\{\left|y_1\right|,\dots \; <!--\; horizontal\; ellipsis\; -->,\left|y_n\right\}\right\}\\ ={\left|x\right|}_{\mathrm{\infty \; <!--\; infinity\; -->}}+{\left|y\right|}_{\mathrm{\infty \; <!--\; infinity\; -->}}\end{array}$

コード(Emacs)

Python 3

#!/usr/bin/env python3
from sympy import pprint, symbols, Matrix, solve
import random
x1, y1, x2, y2, x3, y3 = symbols('x1, y1, x2, y2, x3. y3', real=True)
v1 = Matrix([x1, y1])
v2 = Matrix([x2, y2])
vs = [v1, v2]


def norm1(v):
    return sum([abs(x) for x in v])


def norm2(v):
    return max([abs(x) for x in v])

norms = [norm1, norm2]

print('N1')
for norm in norms:
    try:
        print(all([norm(v) >= 0 for v in vs]))
    except Exception as err:
        print(type(err), err)

for norm in norms:
    for _ in range(5):
        x_sign = -1 if random.randrange(2) == 0 else 1
        y_sign = -1 if random.randrange(2) == 0 else 1
        v = Matrix([x_sign * random.random() * 10,
                    y_sign * random.random() * 10])
        pprint(v)
        pprint(norm(v) >= 0)
print()

print('N2')
for norm in norms:
    try:
        pprint(solve(norm(v1), (x1, y1)))
    except Exception as err:
        print(type(err), err)
    print()
    for v in [(1, 2), (0, 0)]:
        pprint(v)
        pprint(norm(v))
        print()

print('N3')
c = symbols('c', real=True)
for norm in norms:
    try:
        l = norm(c * v1)
        r = abs(c) * norm(v1)
        for t in [l, r, l == r, l.expand() == r.expand(), l.factor() == r.factor()]:
            pprint(t)
            print()
    except Exception as err:
        print(type(err), err)
    print()
    for _ in range(5):
        try:
            x_sign = -1 if random.randrange(2) == 0 else 1
            y_sign = -1 if random.randrange(2) == 0 else 1
            v = Matrix([x_sign * random.random() * 10,
                        y_sign * random.random() * 10])
            l = norm(c * v)
            r = abs(c) * norm(v)
            for t in [l == r, l.expand() == r.expand(), l.factor() == r.factor()]:
                pprint(t)
                print()
        except Exception as err:
            print(type(err), err)
    print()

print('N4')
for nrom in norms:
    try:
        l = norm(v1 + v2)
        r = norm(v1) + norm(v2)
        for t in [l, r, l <= r]:
            pprint(t)
            print()
    except Exception as err:
        print(type(err), err)
    print()
    for _ in range(5):
        try:
            x_sign1 = -1 if random.randrange(2) == 0 else 1
            y_sign1 = -1 if random.randrange(2) == 0 else 1
            v1 = Matrix([x_sign1 * random.random() * 10,
                         y_sign1 * random.random() * 10])
            x_sign2 = -1 if random.randrange(2) == 0 else 1
            y_sign2 = -1 if random.randrange(2) == 0 else 1
            v2 = Matrix([x_sign2 * random.random() * 10,
                         y_sign2 * random.random() * 10])
            l = norm(v1 + v2)
            r = norm(v1) + norm(v2)
            for t in [l, r, l <= r]:
                pprint(t)
                print()
        except Exception as err:
            print(type(err), err)
    print()

入出力結果(Terminal, Jupyter(IPython))

$ ./sample1.py
N1
True
<class 'TypeError'> cannot determine truth value of Relational
⎡-3.60926710185543⎤
⎢                 ⎥
⎣ 6.9103922185622 ⎦
True
⎡9.57041752277116 ⎤
⎢                 ⎥
⎣-3.93028851221304⎦
True
⎡ -7.3944192349977 ⎤
⎢                  ⎥
⎣-0.519473049306546⎦
True
⎡2.38880070572094 ⎤
⎢                 ⎥
⎣-2.03845287170201⎦
True
⎡3.18762488061502⎤
⎢                ⎥
⎣-7.1638944314745⎦
True
⎡-6.41195251111337⎤
⎢                 ⎥
⎣6.05993189592312 ⎦
True
⎡-4.81710291929288⎤
⎢                 ⎥
⎣-5.7781933238635 ⎦
True
⎡ 5.44499970421381 ⎤
⎢                  ⎥
⎣-0.417472228032932⎦
True
⎡8.07251073111729⎤
⎢                ⎥
⎣1.75368978188321⎦
True
⎡3.44930546015265⎤
⎢                ⎥
⎣7.61472240043842⎦
True

N2
[]

(1, 2)
3

(0, 0)
0

<class 'TypeError'> cannot determine truth value of Relational

(1, 2)
2

(0, 0)
0

N3
│c⋅x₁│ + │c⋅y₁│

(│x₁│ + │y₁│)⋅│c│

False

False

False


True

True

True

True

True

True

True

True

True

True

True

True

True

True

True


<class 'TypeError'> cannot determine truth value of Relational

<class 'TypeError'> cannot determine truth value of Relational
<class 'TypeError'> cannot determine truth value of Relational
True

True

True

True

True

True

<class 'TypeError'> cannot determine truth value of Relational

N4
<class 'TypeError'> cannot determine truth value of Relational

14.9446119735654

14.9446119735654

True

17.6787683440230

17.6787683440230

True

12.2781313630320

14.4589230209315

True

14.7469415782085

16.8060896219550

True

6.56786992942470

7.58325266844975

True


6.56786992942470

7.58325266844975

True


7.82660756381296

18.2036912534447

True

7.41865867526981

18.1336184290229

True

9.15529273451527

11.6690643681854

True

5.96971885258877

11.1937075726179

True

18.5451943221697

18.5451943221697

True


$