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barksdalemc
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I tried to expand the [SUM{[X sub k + Y sub k]^2}]^1/2 term but I am stuck there.
Proof of Minkowski Inequality using Cauchy Shwarz
- Thread starter barksdalemc
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- Cauchy Inequality Minkowski Proof
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SUMMARY
The discussion focuses on the proof of the Minkowski Inequality utilizing the Cauchy-Schwarz inequality. The key step involves expanding the term ||\vec{x} + \vec{y}||^2 using the inner product defined in the inner product space, specifically expressed as (\vec{x}+ \vec{y}, \vec{x}+ \vec{y}). By expanding this inner product and applying the Cauchy-Schwarz inequality, the proof is successfully completed. This method eliminates the need for square roots, simplifying the process of proving the Minkowski Inequality.
PREREQUISITES- Understanding of inner product spaces
- Familiarity with the Cauchy-Schwarz inequality
- Knowledge of vector norms and their properties
- Basic proficiency in mathematical proofs
- Study the properties of inner product spaces in detail
- Learn advanced applications of the Cauchy-Schwarz inequality
- Explore the implications of the Minkowski Inequality in functional analysis
- Investigate other inequalities related to norms and inner products
Mathematicians, students studying functional analysis, and anyone interested in the applications of inequalities in vector spaces will benefit from this discussion.
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