Скачать или смотреть Metric Spaces: Definition, Axioms, and Examples

#math #realanalysis #metricspace #manim #visualization #topology #metric #spaces #definition #axioms #examples #euclidean #continuous #functional

In this video, we explore the fundamental concept of a Metric Space. We generalize the intuitive notion of "distance" to rigorous mathematical definitions used in Real Analysis and Topology.

We cover the formal definition, the three key axioms, and visualize three distinct examples of how distance can be measured.

Timeline:
0:05 - Definition of a Metric Space (Set X + Distance Function ρ)
0:48 - The 3 Axioms: Identity, Symmetry, Triangle Inequality
2:43 - Example 1: Euclidean Space (R^n)
3:08 - Example 2: Space of Continuous Functions C[a, b]
3:38 - Example 3: Discrete Metric Space

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