Linear Algebra Done Right · Problem 6A.17

Inner Product Spaces — Sheldon Axler

Mathematics — Linear Algebra · inner-products-and-norms

Problem

Prove that

\left(\sum_{k=1}^{n} a_k b_k\right)^2 \leq \left(\sum_{k=1}^{n} k a_k^2\right)\left(\sum_{k=1}^{n} \frac{b_k^2}{k}\right)

for all real numbers

a_1, \ldots, a_n

and

b_1, \ldots, b_n