Consider a vector space ${\cal S}$ with inner product $(\cdot, \cdot)$. The Cauchy-Schwarz Inequality reads
$$
(y_1, y_1) (y_2, y_2) \geq \left| (y_1, y_2) \right|^2~~\forall y_1, y_2 \in {\cal S}
$$
This inequality is saturated when $y_1 = \lambda y_2$. In particular, this implies
$$
\boxed{ (y_1, y_1) (y_2, y_2) \geq \left| \text{Im} (y_1, y_2) \right|^2 }
$$
My question is the following. Given a fixed $y_1 \in {\cal S}$. Is it always possible to find $y_2 \in {\cal S}$ such that the boxed inequality is saturated?
If not in general, under what conditions is this possible?
PS – I am a physicist, and not that well-versed with math jargon.
Best Answer
You have $|y_1|^2|y_2|^2\geq|(y_1,y_2)|^2\geq|\text{Im}(y_1,y_2)|$. If the right hand side is equal to the left-most hand side then, in particular, you have 'saturation' of Cauchy's inequality. Then $y_2=\lambda y_1$ because the saturation of Cauchy's inequality happens if and only if the vectors are proportional. You then only need to have $|\lambda|^2|y_1|^4=|(y_1,\lambda y_1)|^2=|\text{Im}(\overline{\lambda}(y_1,y_1))|^2=|\text{Im}(\overline{\lambda})|y_1|^2|^2=|\text{Im}(\lambda)|^2|y_1|^4$. This is ok for $\lambda$ imaginary.