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Cairo said:I'm currently reading a book and stuck on an exercise with no solutions.
A proof would be great.
Cairo said:Lp = {a = a(n) | Sigma |a(n)|^p < infinity}.
Thanks in advance.
Cairo said:Thanks Sylas. I can see from the proof that the series is bounded above. But where does the absolute convergence come in?
Also, do I not need to consider finite sums for Holder's Inequality?
Holder's Inequality is a mathematical theorem that states the relationship between two integrals, specifically one that involves two functions raised to a power. It is often used in analysis and functional analysis to prove various theorems and inequalities.
To apply Holder's Inequality, you need to have two functions f and g, and two constants p and q that satisfy the condition 1/p + 1/q = 1. Then, you can use the formula ∫fg ≤ (∫|f|^p)^(1/p) * (∫|g|^q)^(1/q) to prove various inequalities.
Holder's Inequality is an important tool in mathematical analysis, especially in proving various theorems and inequalities involving integrals. It also has applications in other areas such as probability theory and functional analysis.
Yes, Holder's Inequality can be extended to more than two functions. This is known as the generalized Holder's Inequality and it involves using multiple constants and the appropriate conditions to prove the inequality.
Holder's Inequality has various real-life applications, especially in fields such as economics, statistics, and physics. It can be used to prove various inequalities involving integrals, which have practical implications in these fields. It is also used in data analysis and optimization problems.