Brief Summary
Yo, what's up everyone! This lecture is all about metric spaces, focusing on examples like the usual metric space. We'll recap what metric spaces are, go through the definition of the usual metric space, and then dive into proving that it actually satisfies all the required properties. Key points include understanding the four conditions for a metric (non-negativity, equality, symmetry, and triangle inequality) and how they apply to real numbers with the absolute difference as the distance function.
- Metric spaces and their properties are recapped.
- The definition of the usual metric space is explained.
- Proof that the usual metric space satisfies all metric properties.
Introduction
Hello everyone, in this video, we are diving into the second lecture of the Metric Spaces and Complex Analysis series for the 3rd year, 6th-semester mathematics course. We're building upon the previous lecture where we discussed what metric spaces are. In this lecture, we'll be focusing on examples, specifically the usual metric space. If you're new to the channel, make sure to subscribe and hit the bell icon to stay updated.
Recap of Metric Spaces
Before jumping into the usual metric space, let's quickly recap what a metric space is. We have a non-empty set X, and a mapping D defined on that set. This mapping D is called a metric or distance function if it satisfies four conditions: non-negativity, equality condition, symmetric condition, and the triangle inequality. When these conditions are met, the ordered pair (X, D) forms a metric space. Remember, the set and the operation can change the metric space, similar to how groups change in group theory.
Definition of Usual Metric Space
The usual metric space is our first example. It's a metric space that's inherently defined on the set of real numbers (R). Formally, if R is the set of real numbers, and D is a mapping from R x R to R, defined such that d(x, y) = |x - y|, then the ordered pair (R, D) is called the usual metric space. Here, D is referred to as the usual metric. This means the distance between any two real numbers x and y is simply the absolute value of their difference.
Proof: Non-Negativity Condition (M1)
To prove that (R, D) is a metric space, we need to show that D satisfies the four metric conditions. Let's start with the first condition, M1: d(x, y) >= 0 for all x, y in R. We know that d(x, y) = |x - y|. The absolute value of any real number is always non-negative. If x = 0 and y = 0, then |0 - 0| = 0. If x = 2 and y = 1, then |2 - 1| = 1. If x = 1 and y = 2, then |1 - 2| = |-1| = 1. Since the absolute value ensures the result is always greater than or equal to zero, the non-negativity condition is satisfied.
Proof: Equality Condition (M2)
Next, let's prove the equality condition, M2: d(x, y) = 0 if and only if x = y. First, assume d(x, y) = 0. This means |x - y| = 0. The only way the absolute value of a difference can be zero is if the numbers are equal, so x - y = 0, which implies x = y. Conversely, assume x = y. Then x - y = 0, and |x - y| = |0| = 0. Therefore, d(x, y) = 0. Thus, d(x, y) = 0 if and only if x = y.
Proof: Symmetry Condition (M3)
Now, let's prove the symmetry condition, M3: d(x, y) = d(y, x) for all x, y in R. We know that d(x, y) = |x - y|. We can rewrite this as |-(y - x)|. Because the absolute value of a negative number is the same as the absolute value of the positive number, we have |-(y - x)| = |y - x|. And |y - x| is simply d(y, x). Therefore, d(x, y) = d(y, x).
Proof: Triangle Inequality (M4)
Finally, let's prove the triangle inequality, M4: d(x, y) <= d(x, z) + d(z, y) for all x, y, z in R. We start with d(x, y) = |x - y|. We can add and subtract z inside the absolute value: |x - y| = |x - z + z - y|. Now, we use the property that |a + b| <= |a| + |b|. Applying this, we get |x - z + z - y| <= |x - z| + |z - y|. Since |x - z| = d(x, z) and |z - y| = d(z, y), we have d(x, y) <= d(x, z) + d(z, y).
Conclusion
Since D satisfies all four metric conditions, we can conclude that D is a metric, specifically the usual metric. Therefore, the ordered pair (R, D) is a metric space, known as the usual metric space. In the next lecture, we'll explore the Euclidean metric space, another important example. Make sure to revise these concepts, and if you're new to the channel, subscribe and share with your friends. Keep studying, keep striving, and keep succeeding!
