Countable or uncountable number of functions with the given property.

Question

This question was asked in TIFR GS2015 Maths Paper. Let f be a function from {1,2,...,10} to R such that
$$ \left (\sum_{i=1}^{10}\dfrac{|f(i)|}{2^i} \right )^2 = \left (\sum_{i=1}^{10}|f(i)|^2 \right ) \left ( \sum_{i=1}^{10}\dfrac{1}{4^i}\right )$$ Then how many such f are there and are they countable or uncountable? I tried solving it but just couldn't move forward. Any help will be much appreciated.

Answer 1

HINT: Apply the Cauchy-Schwartz inequality; when is it an equality?

Answer 2

By Cauchy Schwarz, equality holds when $(f(1),...,f(10))$ is a multiple of $(2^{-1},..,2^{-10})$.