Metric Spaces Introduction, Real Analysis II

In this lecture, I define the concept of a metric space, a fundamental domain in real analysis. A metric space requires two things: a set M and a function D that measures the distance between elements in M. We'll explore the four essential properties a metric must satisfy: positivity, non-degeneracy, symmetry, and the triangle inequality.

(MA 426 Real Analysis II, Lecture 4)

We begin by defining these properties in detail. Positivity means the distance is never negative. Non-degeneracy ensures the distance is zero if and only if the elements are the same. Symmetry indicates that the distance from X to Y is the same as from Y to X. The triangle inequality states that the direct distance between two points is always less than or equal to the distance traveled via a third point.

To illustrate, we explore a discrete metric space, where the distance as either 0 or 1. We'll verify this space meets all metric properties. Next, we examine the taxi cab metric on the xy-plane, calculating distances based on city block paths rather than straight lines. We'll check that this metric also satisfies the four properties and discuss its unique geometric implications, such as diamond-shaped circles.

Finally, I demonstrate that any normed vector space induces a metric, proving that the norm's properties ensure the metric properties. This concludes with a discussion on the broader implications and limitations of metrics derived from norms.

#advancedcalculus #MathLecture #MetricSpace #RealAnalysis #Mathematics #MathProof #DiscreteMetric #TaxiCabMetric #EuclideanSpace #Geometry #NormedVectorSpace #MathConcepts

[PaulHuaranca]

Thanks for the clear explanation! Really helped me understand. 😃

[dulce_de_leche]

i have a quiz on metric spaces tomorrow and finding this explanation was like a breath of fresh air - so much clearer!! thank you!!

[Ezio_Beta]

Hi! I just discovered your channel and I love it. I also used to be in academia and have a real passion for teaching. Keep up the great work, and thank you for helping make quality education accessible to everyone.

[schweinsian]

Hi mam
Hoping having good health...
Lectures of metric space are best .

[NamanyaIsaac-zi]

Thanks so much, but I would like to get the whole course unit of topology

[eswyatt]

In a way, taxicab distance is why I had such a problem with Gram/Schmidt. How can this angled ray from the origin be the sum of the distance in the x-direction and the distance in the y-direction. It seemed to violate the pythagorean theorem. But that's not what it was doing. It was adding vector components, not distances!

[zidfor-hode]

Hello teacher, will you do a course about real analysis I ?