Prove that limp →∞𝐯p=𝐯∞ for all 𝐯 ∈ℝ^2.

Name: Problem 16, Chapter 3: Applied Linear Algebra (Undergraduate Texts in Mathematics)
Uploaded: 2022-02-07T07:58:13-08:00
Duration: 7 min 46 s
Channel: Shafiq Rehman
Description: Prove that limp →∞𝐯p=𝐯∞ for all 𝐯 ∈ℝ^2.

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Key Concepts

p-norm

The p-norm is a way to measure the magnitude of vectors by raising the absolute values of each component to the power p, summing these values, and then taking the p-th root of the sum. It generalizes the concept of Euclidean distance (which corresponds to p = 2) to any real number p ? 1, helping quantify sizes and distances in vector spaces.

Infinity norm

The infinity norm of a vector is defined as the largest absolute value among its components. It provides a measure of the vector’s magnitude by focusing solely on its largest entry, offering an alternative perspective to norms that aggregate contributions from all components.

Convergence of norms

The proof that the limit of the p-norm as p approaches infinity equals the infinity norm relies on the idea that, for large p, the largest component of the vector dominates the sum in the p-norm. This reflects a general concept in analysis where, under certain limits, a particular term or factor can overpower all others, leading to convergence toward a simpler form of measurement.

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