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Let $f$ be a function from $\left \{ 1, 2,....10 \right \}$ to $\mathbb{R}$ such that 

$(\sum_{i=1}^{10}\frac{|f(i)|}{2^{i}})^{2} = (\sum_{i=1}^{10} |f(i)|^{2}) (\sum_{i=1}^{10}\frac{1}{4^{i}})$

Make the correct statement.

  1. There are uncountably many $f$ with this property.
  2. There are only countably infinitely many $f$ with this property.
  3. There is exactly one such $f$
  4. There is no such $f$.
asked in Set Theory & Algebra by Boss (29.5k points) | 75 views
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Option D can not be the ans, because when f(i)=0 for all i=1 to 10 then it satisfies the equation. So at least one solution is possible.

But how to get exact solution ??

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