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.
- There are uncountably many $f$ with this property.
- There are only countably infinitely many $f$ with this property.
- There is exactly one such $f$
- There is no such $f$.