A sum in Hilbert space

Question

Let $\mathbb R^n$ be the Hilbert space ($n\in\mathbb N^*$) with the following scalar product

$$\langle x,y\rangle= \sum_{i=1}^n x_i y_i$$

Let $a_1,a_2,...,a_n \in \mathbb R_+^*$ verifying

$$\sum_{i=1}^n a_i =1$$

How do I show that

$$\sum_{i=1}^n \frac{1}{a_i}\ge n^2$$

and when do I have equality ?

Answer 1

$n=\sum \frac 1 {\sqrt a_i} \sqrt {a_i} \leq \sqrt {\sum a_i} \sqrt {\sum \frac 1 {a_i}}$ by Cauchy -Schwarz inequality. Now square both sides.

Answer 2

Apply the AM-GM inequality to both “factors” of the product: $(a_1+a_2+....+a_n)(\frac{1}{a_1}+\frac{1}{a_2}+...+\frac{1}{a_n}) \ge n^2$. From this we have the desire inequality since $a_1+a_2+...+a_n = 1$ and $ = $ occurs when $a_1 = a_2 = ...= a_n = \frac{1}{n} $ .