Real Analysis: Spaces, Limits, and Continuity

This lecture covers the fundamental concepts in Real Analysis, including spaces, limits, and continuity. It starts with a review of prerequisites from Analysis I and Linear Algebra, then delves into topics such as vector spaces, norms, and product spaces. The lecture also explores the definition of sequences, convergence, and the continuity of functions in R^n. Key notions like Cauchy sequences, completeness, and the Bolzano-Weierstrass theorem are discussed, leading to the understanding of open and closed sets, convergence, and continuity in real spaces.