Does Hölder's Inequality implies that $\int_{-\infty}^\infty |f(t) \,g(t)|\,\mathrm dt \leq \int_{-\infty}^\infty |f(t)|\,\mathrm dt \, \sup |g|$?

Question

Does Hölder's Inequality implies that $\int_{-\infty}^\infty |f(t) \,g(t)|\,\mathrm dt \leq \int_{-\infty}^\infty |f(t)|\,\mathrm dt \sup_t |g(t)|$ ?

I am thinking in the case $p=1$ and $q=\infty$ so $1=\frac{1}{p}+\frac{1}{q}$ holds, but I am not sure if the norms $||\cdot||_\infty$ and $||\cdot||_1$ are well interpreted from what it is said on Wikipedia here and here, and also if the inequality is valid on unbounded domains as $(-\infty;\,\infty)$.

Also please this related question I had left here. Thanks you very much.

Answer 1

Yes, it works ... and the proof is quite immediate: just notice that $|f(t)\,g(t)|≤ (\sup_s{|g(s)|})\ |f(t)|$, and now since $\sup_s{|g(s)|}$ is just a constant independent of $t$, you can put it out of the integral ...