Cauchy-Schwarz Inequality for Integrals for any two functions clarification

Cauchy-Schwarz says
∫Efgdx≤(∫Ef2dx)1/2(∫Eg2dx)1/2\int_{E} f g \mathrm{d} x \leq\left(\int_{E} f^{2} \mathrm{d} x\right)^{1 / 2}\left(\int_{E} g^{2} \mathrm{d} x\right)^{1 / 2}∫Efgdx≤(∫Ef2dx)1/2(∫Eg2dx)1/2
where ∫E\int_{E}∫E is a definite integral. Then the AM−GM‾\underline{\mathrm{AM}-\mathrm{GM}}AM−GM says that for a,b≥0a, b \geq 0a,b≥0
ab≤a+b2\sqrt{a b} \leq \frac{a+b}{2} ab≤2a+b
Applying (2) to (1) yields (for c>0)(\text { for } c>0)( for c>0)
∫Efgdx=∫E(cf)(g/c)dx≤c2∫Ef2dx+12c∫Eg2dx\begin{aligned} \int_{E} f g \mathrm{d} x &=\int_{E}(\sqrt{c} f)(g / \sqrt{c}) \mathrm{d} x \\ & \leq \frac{c}{2} \int_{E} f^{2} \mathrm{d} x+\frac{1}{2 c} \int_{E} g^{2} \mathrm{d} x \end{aligned}∫Efgdx=∫E(cf)(g/c)dx≤2c∫Ef2dx+2c1∫Eg2dx

LINK: https://math.stackexchange.com/questions/351905/cauchy-schwarz-inequality-for-integrals-for-any-two-functions-clarification

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