Functional Analysis 2 | Examples for Metrics - Euclidean or Discrete Metric? [dark version]

TL;DR

A metric space is a set X equipped with a distance function that satisfies positive definiteness, symmetry, and the triangle inequality.

Briefing Cornell Notes

Briefing

Metric spaces aren’t just a formal definition—they come with concrete distance rules that change what “closeness” and “circles” look like. After introducing the basic framework (a set X equipped with a metric), the discussion drills into three key examples on which later ideas like limits and accumulation points will rely.

On the complex numbers, the metric is the usual distance given by the absolute value: for points x and y in ℂ, the distance is |x − y|. Geometrically, this matches the standard picture in the complex plane: subtracting two complex numbers gives a vector, and its length is exactly the distance between the original points. The same idea extends to higher dimensions. For ℝ^n, the Euclidean metric measures distance using the square root of the sum of squared coordinate differences—d(x, y) = √((x1 − y1)^2 + (x2 − y2)^2 + … + (xn − yn)^2). This choice is tied to familiar geometry (including Pythagoras’ theorem), and it’s presented as the most common but not the only possible metric.

A second example replaces Euclidean distance with a “maximum coordinate difference” rule, producing the metric d(x, y) = max_i |xi − yi| on ℝ^n. The key geometric takeaway is that the shapes of “equal distance” sets change dramatically. Under the Euclidean metric, points at a fixed distance from x form a circle (or sphere in higher dimensions). Under the max metric, the corresponding “circle” can look very different because the distance is determined by the largest coordinate discrepancy. Visualizing the component-wise differences clarifies why: the overall distance is whichever coordinate difference is biggest, not the combined effect of all coordinates.

The final example is more abstract and works on any nonempty set X: the discrete metric. It defines distance by a simple rule—d(x, y) = 0 when x = y, and d(x, y) = 1 when x ≠ y. The discussion emphasizes that the metric axioms are straightforward here: symmetry is automatic from the definition, and the triangle inequality is checked by splitting into cases. If x = y, then d(x, y) = 0 and the inequality holds because distances are never negative. If x ≠ y, then d(x, y) = 1, and the intermediate point z can only contribute distances of 0 or 1; the sum on the right can be at most 2, so the triangle inequality still holds. The resulting geometry is “all isolated points”: every distinct pair of points stays at the same positive distance, so there are no genuine neighbors.

Together, these examples show how changing the metric reshapes the geometry of a space—sometimes in subtle ways (Euclidean vs. max), sometimes completely (discrete metric). That flexibility is exactly why metric spaces are powerful in functional analysis: the same underlying set can support very different notions of convergence and proximity depending on the distance rule.

Cornell Notes

Metric spaces are built by putting a distance rule on a set X, and the choice of metric determines what “closeness” means. The complex numbers use the distance |x − y|, matching the usual geometry in the complex plane. On ℝ^n, the Euclidean metric uses the square root of the sum of squared coordinate differences, while the max metric uses the largest absolute coordinate difference, producing very different “circles” (level sets of constant distance). For any nonempty set X, the discrete metric sets d(x, y)=0 if x=y and d(x, y)=1 otherwise; it satisfies the triangle inequality by a two-case check and yields isolated points with no neighbors. These examples illustrate how geometry—and later concepts like convergence—depend on the metric.

Why does the metric on ℂ given by |x − y| match the usual notion of distance?

For complex numbers x and y, the difference x − y is another complex number whose modulus |x − y| represents the length of the corresponding vector in the complex plane. That length is exactly the distance between the points x and y, so the metric reproduces standard planar geometry.

How do the Euclidean and max metrics differ in what determines distance?

The Euclidean metric combines all coordinate differences: it squares each (xi − yi), sums them, and then takes the square root. The max metric ignores the combined effect and instead takes the single largest coordinate discrepancy: d(x, y) = max_i |xi − yi|. As a result, the max metric is controlled by whichever coordinate differs most.

What geometric change should be expected when switching from Euclidean distance to the max metric?

Under Euclidean distance, points at a fixed distance from a center form a circle (or sphere in higher dimensions). With the max metric, the set of points at a fixed distance can look completely different because the distance is determined by the maximum coordinate difference. So “circles” become shapes defined by the coordinate-wise constraint rather than by Pythagorean geometry.

How does the discrete metric guarantee the triangle inequality?

The discrete metric assigns only two possible distances: 0 when x=y and 1 when x≠y. For the triangle inequality d(x, y) ≤ d(x, z) + d(z, y): if x=y, then d(x, y)=0 and the inequality holds since the right side is nonnegative. If x≠y, then d(x, y)=1; meanwhile d(x, z) and d(z, y) can only be 0 or 1, so their sum is at most 2, and the inequality still holds.

What does the discrete metric imply about “neighbors” in the space?

Because every distinct point is always at distance 1 from a given point, there are no points arbitrarily close to it. That makes every point isolated: there are no genuine neighbors in the usual metric sense, since the distance jumps immediately from 0 (same point) to 1 (any other point).

Review Questions

Given x, y ∈ ℝ^n, compute the Euclidean distance and the max-metric distance for a specific pair of vectors and compare which coordinate differences control the result.
Explain why the level set {z : d(x, z)=r} looks like a circle under the Euclidean metric but can look different under the max metric.
Prove the triangle inequality for the discrete metric by explicitly handling the cases x=y and x≠y.

Key Points

1
A metric space is a set X equipped with a distance function that satisfies positive definiteness, symmetry, and the triangle inequality.
2
On ℂ, the standard metric is d(x, y)=|x−y|, matching the usual geometry of the complex plane.
3
On ℝ^n, the Euclidean metric uses d(x, y)=√(∑_{i=1}^n (xi−yi)^2), reflecting Pythagorean geometry.
4
The max metric on ℝ^n is d(x, y)=max_i |xi−yi|, so distance is determined by the single largest coordinate difference.
5
Changing the metric changes the shape of “equal-distance” sets: Euclidean circles can become very different under the max metric.
6
The discrete metric d(x, y)=0 if x=y and d(x, y)=1 otherwise works on any nonempty set and yields isolated points with no neighbors.
7
The triangle inequality for the discrete metric follows from the fact that all distances are only 0 or 1, enabling a simple two-case check.

Highlights

Euclidean distance blends all coordinate differences via squares and a square root, while the max metric picks out only the largest coordinate discrepancy.

Under the max metric, “circles” (constant-distance sets) can look completely different from Euclidean circles because the distance ignores the smaller coordinate differences.

The discrete metric turns any nonempty set into a space where every distinct pair of points sits at the same distance 1, making all points isolated.

The triangle inequality for the discrete metric is verified by splitting into x=y (distance 0) and x≠y (distances only 0 or 1).

Topics

Metric Spaces
Euclidean Metric
Max Metric
Discrete Metric
Triangle Inequality