Duality product - Inequality

Question

If I define the duality product between $L^2(\Omega)\times H^{-1}(\Omega)$ and $L^2(\Omega)\times H^1_0(\Omega)$ as follows:

for all $(\phi_0,\phi_1)\in L^2(\Omega)\times H^{-1}(\Omega)$ and $(f_0,f_1)\in L^2(\Omega)\times H^1_0(\Omega).$

$$<(\phi_0,\phi_1),(f_0,f_1)>=<\phi_1,f_1>_{H^1_0,H^{-1}}-\int_{\Omega}\phi_0 f_0 dx$$

I was wondering if we have the following inequality:

$$<(\phi_0,\phi_1),(f_0,f_1)>\leq ||(\phi_0,\phi_1)||_1||(f_0,f_1)||_2$$ where $||.||_1 $ and $||.||_2 $ are the norms in the product respective spaces.

I applied Cauchy-Scwartz inequality plus the fact that: $$ab\leq a^2+b^2$$ And I got $$<(\phi_0,\phi_1),(f_0,f_1)>\leq ||(\phi_0,\phi_1)||_1^2+||(f_0,f_1)||_2^2$$

Answer 1

Hint:

For real numbers $a,b,c,d$, show that the inequality $$ac+bd\leq \sqrt{a^2+b^2}\sqrt{c^2+d^2}$$ holds. Then use that inequality appropriately.