数学 - Python - JavaScript - 解析学 - 連続写像の空間 - ノルム空間(定義、閉区間上で定義された実連続関数全体が作るベクトル空間、完備性、関数列、コーシー列、極限、収束・発散)

学習環境

Surface 3 (4G LTE)、Surface 3 タイプカバー、Surface ペン(端末)
Windows 10 Pro (OS)
Nebo(Windows アプリ)
iPad Pro + Apple Pencil
MyScript Nebo(iPad アプリ)
参考書籍

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2.を取り組んでみる。

1. ${∥f∥}_{1} = \int_{0}^{1} |f (x)| dx \geq 0$
2. ${∥f∥}_{1} = \int_{0}^{1} |f (x)| dx = 0 \Rightarrow |f (x)| = 0$
  
  よって
  
  $\forall x \in [0, 1] [f (x) = 0]$
  
  ゆえに、
  
  ${∥f∥}_{1} = 0 \Rightarrow \forall x \in [0, 1] [f (x) = 0]$
3. $\begin{array}{l} | | c f | |_{1} = \int_{0}^{1} |c f (x)| dx \\ = |c| \int_{0}^{1} |f (x)| dx \\ = |c| {∥f∥}_{1} \end{array}$
4. $\begin{array}{l} {∥f + g∥}_{1} \\ = \int_{0}^{1} |(f + g) (x)| dx \\ = \int_{0}^{1} |f (x) + g (x)| dx \\ \leq \int_{0}^{1} (|f (x)| + |g (x)|) dx \\ = \int_{0}^{1} |f (x)| dx + \int_{0}^{1} |g (x)| dx \\ = {∥f∥}_{1} + {∥g∥}_{1} \end{array}$
よって V 上のノルムである

関数列、

$f_{n} (x) = {\begin{cases} n x (0 \leq x \leq \frac{1}{n}) \\ 1 (x > \frac{1}{n}) \end{cases}$

はコーシー列である。

この関数列の極限を考えると、 V の中で収束しない。

よって、 V は完備ではない。

コード(Emacs)

Python 3

#!/usr/bin/env python3
from sympy import pprint, symbols, Limit, oo

x = symbols('x')
n = symbols('n', integer=True)

f1 = n * x
f2 = 1

l1 = Limit(f1, n, oo)
l2 = Limit(f2, n, oo)

g1 = l1.doit()
g2 = l2.doit()

d = {x: 0}

for t in [l1, l2, g1, g2, g1.subs(d), g2.subs(d)]:
    pprint(t)
    print()

入出力結果(Terminal, Jupyter(IPython))

$ ./sample2.py
lim (n⋅x)
n─→∞     

lim 1
n─→∞ 

∞⋅sign(x)

1

nan

1

$

HTML5

<div id="graph0"></div>
<pre id="output0"></pre>
<label for="r0">r = </label>
<input id="r0" type="number" min="0" value="0.5">
<label for="dx">dx = </label>
<input id="dx" type="number" min="0" step="0.0001" value="0.0005">
<br>
<label for="x1">x1 = </label>
<input id="x1" type="number" value="0">
<label for="x2">x2 = </label>
<input id="x2" type="number" value="1">
<br>
<label for="y1">y1 = </label>
<input id="y1" type="number" value="0">
<label for="y2">y2 = </label>
<input id="y2" type="number" value="1">
<br>

<label for="n0">n0 = </label>
<input id="n0" type="number" min="1" step="1" value="6">

<button id="draw0">draw</button>
<button id="clear0">clear</button>

<script type="text/javascript" src="https://cdnjs.cloudflare.com/ajax/libs/d3/4.2.6/d3.min.js" integrity="sha256-5idA201uSwHAROtCops7codXJ0vja+6wbBrZdQ6ETQc=" crossorigin="anonymous"></script>

<script src="sample2.js"></script>

JavaScript

let div0 = document.querySelector('#graph0'),
    pre0 = document.querySelector('#output0'),
    width = 600,
    height = 600,
    padding = 50,
    btn0 = document.querySelector('#draw0'),
    btn1 = document.querySelector('#clear0'),
    input_r = document.querySelector('#r0'),
    input_dx = document.querySelector('#dx'),
    input_x1 = document.querySelector('#x1'),
    input_x2 = document.querySelector('#x2'),
    input_y1 = document.querySelector('#y1'),
    input_y2 = document.querySelector('#y2'),
    input_n0 = document.querySelector('#n0'),
    inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2,
              input_n0],
    p = (x) => pre0.textContent += x + '\n',
    range = (start, end, step=1) => {
        let res = [];
        for (let i = start; i < end; i += step) {
            res.push(i);
        }
        return res;
    };

let f = (n) => (x) => 0 <= x && x <= 1 / n ? n * x : 1,
    f1 = f(1),
    f2 = f(2),
    f3 = f(3),
    f4 = f(4),
    f5 = f(5);


let draw = () => {
    pre0.textContent = '';


    let r = parseFloat(input_r.value),
        dx = parseFloat(input_dx.value),
        x1 = parseFloat(input_x1.value),
        x2 = parseFloat(input_x2.value),
        y1 = parseFloat(input_y1.value),
        y2 = parseFloat(input_y2.value),
        n0 = parseInt(input_n0.value, 10);
    
    if (r === 0 || dx === 0 || x1 > x2 || y1 > y2) {
        return;
    }

    let points = [],
        lines = [],
        fn = f(n0),
        fns = [[f1, 'red'],
               [f2, 'green'],
               [f3, 'blue'],
               [f4, 'orange'],
               [f5, 'brown'],
               [fn, 'skyblue']];

    fns
        .forEach((o) => {
            let [fn, color] = o;
            
            for (let x = x1; x <= x2; x += dx) {
                let y = fn(x);
                
                if (Math.abs(y) < Infinity) {
                    points.push([x, y, color]);
                }
            }
        });
    
    let xscale = d3.scaleLinear()
        .domain([x1, x2])
        .range([padding, width - padding]);

    let yscale = d3.scaleLinear()
        .domain([y1, y2])
        .range([height - padding, padding]);

    let xaxis = d3.axisBottom().scale(xscale);
    let yaxis = d3.axisLeft().scale(yscale);
    div0.innerHTML = '';
    let svg = d3.select('#graph0')
        .append('svg')
        .attr('width', width)
        .attr('height', height);

    svg.selectAll('circle')
        .data(points)
        .enter()
        .append('circle')
        .attr('cx', (d) => xscale(d[0]))
        .attr('cy', (d) => yscale(d[1]))
        .attr('r', r)
        .attr('fill', (d) => d[2] || 'green');

    svg.selectAll('line')
        .data([[x1, 0, x2, 0], [0, y1, 0, y2]].concat(lines))
        .enter()
        .append('line')
        .attr('x1', (d) => xscale(d[0]))
        .attr('y1', (d) => yscale(d[1]))
        .attr('x2', (d) => xscale(d[2]))
        .attr('y2', (d) => yscale(d[3]))
        .attr('stroke', (d) => d[4] || 'black');
    
    svg.append('g')
        .attr('transform', `translate(0, ${height - padding})`)
        .call(xaxis);

    svg.append('g')
        .attr('transform', `translate(${padding}, 0)`)
        .call(yaxis);
    p(fns.join('\n'));
};

inputs.forEach((input) => input.onchange = draw);
btn0.onclick = draw;
btn1.onclick = () => pre0.textContent = '';
draw();

r = dx =
x1 = x2 =
y1 = y2 =
n0 =

Kamimura's blog

2017年12月19日火曜日

数学 - Python - JavaScript - 解析学 - 連続写像の空間 - ノルム空間(定義、閉区間上で定義された実連続関数全体が作るベクトル空間、完備性、関数列、コーシー列、極限、収束・発散)

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