Distance

1.1 Distance

Linear space $R^{m} = {x = (x_{1}, . . ., x_{m})^{T} | x_{1}, . . ., x_{m} \in R}$ . Define the addition and number multiplication on the linear space, $\begin{aligned} x + y & = (x_{1} + y_{1}, . . ., x_{m} + y_{m})^{T} \\ λ x & = (λ x_{1}, . . ., λ x_{m})^{T} \end{aligned}$

Why should it be written as a column vector? Let $\begin{aligned} z & = 2 x + 3 y = f (x, y) \\ = (\begin{array}{c} 2 & 3 \end{array}) (\begin{array}{c} x \\ y \end{array}) \end{aligned}$

The former row vector is called a function, the latter column vector is called a vector (or point).

Definition 1.1.1 The distance on $R^{m}$ is a function $d : R^{m} \times R^{m} \to R$ such that

1.: For any $x, y$ , $d (x, y) = d (y, x) \geq 0$ .
2.: $d (x, y) = 0 \Leftrightarrow x = y$ .
3.: For any $x, y, z$ , $d (x, y) \leq d (x, z) + d (z, y)$ .

Moreover, if for any $x, y, z$ , $d (x, y) = d (x + z, y + z)$ , then $d$ is translational invariant.

Definition 1.1.2 The norm on $R^{m}$ is a function $∥ \cdot ∥ : R^{m} \to R$ such that

1.: For any $x$ , $∥ x ∥ \geq 0$ and $∥ x ∥ = 0 \Leftrightarrow x = 0$ .
2.: $∥ λ x ∥ = | λ | ∥ x ∥$ .
3.: $∥ x + y ∥ \leq ∥ x ∥ + ∥ y ∥$ .

Corollary 1.1.3 If $∥ \cdot ∥$ is the norm on $R^{m}$ , then let $d (x, y) = ∥ x - y ∥$ , here $d$ is a translational invariant distance.

Definition 1.1.4 The inner product on $R^{m}$ is a binary function $⟨ \cdot, \cdot ⟩ : R^{m} \times R^{m} \to R$ such that

1.: $⟨ x, y ⟩ = ⟨ y, x ⟩$ .
2.: $⟨ x, x ⟩ \geq 0$ , $⟨ x, x ⟩ = 0 \Leftrightarrow x = 0$ .
3.: Binary linear. $⟨ α x + β y, z ⟩ = α ⟨ x, z ⟩ + β ⟨ y, z ⟩$ 。

Corollary 1.1.5 If $⟨ \cdot, \cdot ⟩$ is the inner product on $R^{m}$ , then let $∥ x ∥ = \sqrt{⟨ x, x ⟩}$ , here $⟨ \cdot, \cdot ⟩$ is a norm which subjects to the Cauchy-Schwartz inequality $⟨ x, y ⟩ \leq ∥ x ∥ ∥ y ∥ = \sqrt{⟨ x, x ⟩ ⟨ y, y ⟩}$ .

Example 1.1.1 Standard inner product on $R^{m}$ is defined as $⟨ x, y ⟩ = \sum_{i = 1}^{m} x_{i} y_{i}$ . $e_{1}, . . ., e_{m}$ are called the identity orthogonal bases. Define the norm $∥ x ∥ = \sqrt{\sum_{i = 1}^{m} x_{i}^{2}}$ and the distance $d (x, y) = \sqrt{\sum_{i = 1}^{m} (x_{i} - y_{i})^{2}}$ . They compose a Euclid space. Generally, for any given $p \geq 1$ , define the distane $d_{p} (x, y) = (\sum_{i = 1}^{m} | x_{i} - y_{i} |^{p})^{\frac{1}{p}}$ and the norm $∥ x ∥_{p} = (\sum_{i = 1}^{m} | x_{i} |^{p})^{\frac{1}{p}}$ . $∥ \cdot ∥_{p}$ is consistent with a inner product if and only if $p = 2$ .

When $p \to + \infty$ , $∥ x ∥_{\infty} = max_{1 \leq i \leq m} | x_{i} |$ and $d (x, y)_{\infty} = max_{1 \leq i \leq m} | x_{i} - y_{i} |$ .