Introduction to inner product, norm, and metric on Rn, Real Analysis II
In this video, we introduce second-semester real analysis (aka Advanced Calculus II) and start our exploration of the spaces we will study throughout the course. In this video, we start with Euclidean spaces R^n (such as the real line R, the plane R^2, and three-dimensional space R^3).
(MA 426 Real Analysis II, Lecture 1)
We cover key concepts including vector spaces, inner products, norms, and metrics, explaining each in the context of Euclidean space. We illustrate how these concepts relate to geometric ideas like distance, length, and angle. Additionally, we provide a brief overview of vector notation and operations, emphasizing the importance of understanding vectors as both points and directions in space.
Lastly we work through a couple of exercises on norms, proving statements about vector size and coordinate bounds. We can now use these bounds whenever we need them in the course.
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