Example of metric space other than Euclidean space

A metric space is a set equipped with a metric (a function defining the distance between elements). The Euclidean space is a common example, but here are some other examples:

### 1. Discrete Metric Space
**Definition:** For any set \(X\), the discrete metric \(d\) is defined as:
\[
d(x, y) =
\begin{cases}
0 & \text{if } x = y, \\
1 & \text{if } x \neq y.
\end{cases}
\]
**Example:** Consider the set \(X = \{a, b, c\}\). The distance between any two distinct elements is 1, and the distance from any element to itself is 0.

### 2. Taxicab (Manhattan) Space
**Definition:** In \( \mathbb{R}^n \), the taxicab (or Manhattan) metric is defined as:
\[
d((x_1, x_2, \dots, x_n), (y_1, y_2, \dots, y_n)) = \sum_{i=1}^n |x_i - y_i|.
\]
**Example:** In \( \mathbb{R}^2 \), the distance between points \( (1, 2) \) and \( (4, 6) \) is \( |1 - 4| + |2 - 6| = 3 + 4 = 7 \).

### 3. Maximum Metric Space (Chebyshev Space)
**Definition:** In \( \mathbb{R}^n \), the maximum metric (or Chebyshev distance) is defined as:
\[
d((x_1, x_2, \dots, x_n), (y_1, y_2, \dots, y_n)) = \max_{1 \leq i \leq n} |x_i - y_i|.
\]
**Example:** In \( \mathbb{R}^2 \), the distance between \( (1, 2) \) and \( (4, 6) \) is \( \max(|1 - 4|, |2 - 6|) = \max(3, 4) = 4 \).

### 4. Sequence Space \( \ell^p \)
**Definition:** The space of all infinite sequences of real numbers \( x = (x_1, x_2, \dots) \) with the \( p \)-norm:
\[
d(x, y) = \left(\sum_{i=1}^{\infty} |x_i - y_i|^p \right)^{1/p},
\]
for \( 1 \leq p < \infty \).
**Example:** Consider the sequence space \( \ell^2 \) (the space of square-summable sequences), where the distance is the square root of the sum of the squares of the differences.

### 5. The Space of Continuous Functions
**Definition:** Let \( C([a, b]) \) denote the set of continuous functions on the interval \( [a, b] \) with the metric:
\[
d(f, g) = \max_{x \in [a, b]} |f(x) - g(x)|.
\]
**Example:** Consider \( C([0, 1]) \) with \( f(x) = x \) and \( g(x) = x^2 \). The distance between \( f \) and \( g \) is \( \max_{x \in [0, 1]} |x - x^2| = \frac{1}{4} \).

Would you like more details or have any questions?

### Related Questions:
1. What are the properties that a metric must satisfy?
2. How is the Manhattan metric used in practical applications?
3. Can you explain the concept of convergence in different metric spaces?
4. What is the significance of the \( \ell^p \) spaces in functional analysis?
5. How does the maximum metric compare to the Euclidean metric?

**Tip:** Always verify that the function you are using to define distances satisfies all the properties of a metric: non-negativity, identity of indiscernibles, symmetry, and the triangle inequality.

Explore various examples of metric spaces beyond Euclidean space, including discrete metric space, taxicab (Manhattan) space, maximum metric space (Chebyshev space), sequence space ℓ^p, and the space of continuous functions. Understand their definitions, key examples, and applications.

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