Cardinality of the set of functions which holds the Equality

Question

Let $f$ be a function from $\{1,2,3, \dots ,10 \}$ to $\mathbb R$ such that $$\bigg( \sum_{i=1}^{10}\frac{|f(i)|}{2^i}\bigg)^2 = \bigg( \sum_{i=1}^{10} |f(i)|^2 \bigg) \bigg(\sum_{i=1}^{10} \frac{1}{4^i} \bigg)$$

Let $S$ be the set of all functions which holds this Equality. Find the Cardiality of $S$

I have no idea how to start . I would be thankful if someone help me.

Answer 1

This is really Cauchy-Schwarz inequality, which says that

$$\left(\sum_{i=1}^n a_i b_i\right)^2 \le \left(\sum_{i=1}^n a_i^2 \right) \left( \sum_{i=1}^n b_i^2 \right)$$

and equality holds if and only if $a_i = c b_i$ for some $c\in \mathbb R$. Now in your situation, $a_i = |f(i)|$, $b_i = 1/2^i$, which means that you must have

$$ |f(i)| = c\frac{1}{2^i},\ \ i=1, 2, \cdots, 10$$

for some $c\in \mathbb R$. In particular, the cardinatlity of $S$ is that of $\mathbb R$.