Hölder's Inequality Proof using Young's Inequality | Functional Analysis

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In this video, we present a complete proof of Hölder's Inequality for vectors in Euclidean space (R^n).

We start by stating the inequality and the condition of conjugate exponents (1/p + 1/q = 1). The proof is broken down into three logical steps:
1. Using the Homogeneity property to normalize the vectors.
2. Proving Young's Inequality geometrically using the curve y = x^(p-1) and comparing areas.
3. Summing the results to derive the final inequality.

Timeline:
0:00 - Statement of Hölder's Inequality
1:10 - Proof Strategy: Homogeneity & Normalization
2:05 - Geometric Proof of Young's Inequality
3:27 - Substituting Areas & Algebraic Summation
4:30 - Final Derivation & Conclusion

This result is a fundamental tool in Functional Analysis and Measure Theory, generalizing the famous Cauchy-Schwarz inequality.

Visualization created using Python and the Manim library.


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