Re: [分析] 內積與距離, 畢式定理

[cgkm (cgkm)]

※ 引述《GSXSP (Gloria)》之銘言：
: : --
: : ◆ From: 140.113.211.193
: : → Xixan       :沒有內積  怎麼會有直角?                            11/19 20:26
: : → GSXSP       :有定義內積阿 我是指距離的定義方式不是用內積的型式  11/19 21:44

Pythagorean theorem for an inner product space V:
        If ∥‧∥ is the norm induced by the inner product
        and u and v in V are orthogonal (with respect to the
        inner product), then
                     2         2            2
                ∥u∥  +  ∥v∥  = ∥u + v∥ .

If you try to use a different norm, this obviously may not hold.

Example:

   y
 1 │
   │＼
   │  ＼
   │    ＼
   ┼──── x
   0        1

           2
Consider Ｒ  with the norm

        (x, y) |→ ∥(x, y)∥ = max{|x|, |y|}.

The points (0, 1) and (1, 0) are orthognal with respect to
the Euclidean inner product and

                  2              2
        ∥(0, 1)∥  +  ∥(1, 0)∥  = 2,

but this is not equal to
                           2              2
        ∥(0, 1) - (1, 0)∥  = ∥(-1, 1)∥  = 1.

--
※ 發信站: 批踢踢實業坊(ptt.cc)
◆ From: 75.62.141.216
→ GSXSP       :我知道不一定成立  我的問題是有沒有可能成立
→ GSXSP       :不知道是不是問得有點模糊@@
                                2
Claim. If ∥‧∥ is a norm on Ｒ  such that
             2         2            2
        ∥u∥  +  ∥v∥  = ∥u + v∥
                   2
whenever u, v in Ｒ  are perpendicular (i.e. orthogonal
in the Euclidean sense), then
                          2     2
        ∥(x, y)∥ = √(ax  + by )

where
                      2                      2
        a = ∥(1, 0)∥   and   b = ∥(0, 1)∥ .

If a = b, then ∥‧∥ is a constant multiple of the Euclidean norm.

                          2
Proof. Let (x, y) be in Ｒ . Then (x, 0) and (0, y) are perpendicular,
so
                  2                      2
        ∥(x, y)∥  = ∥(x, 0) + (0, y)∥
                                2             2
                    = ∥(x, 0)∥  + ∥(0, y)∥
                       2          2    2          2
                    = x ∥(1, 0)∥  + y ∥(0, 1)∥
                        2     2
                    = ax  + by.     □
※ 編輯: cgkm            來自: 75.62.141.216        (11/20 14:52)
推 math1209    :of course, 有可能成立. 但這到底成不成立都不重要.
→ math1209    :比較有意義反而是你之後的逆敘述...
→ math1209    :上兩行回應 GSXSP.
推 math1209    :我關心的是:無限維度...=.=
→ math1209    :我傾向於對於 GSXSP 之後的那個逆敘述, 那兩個
→ math1209    :norms 不一定等價. (有限維度在此就變成 non-sense)