0

Prove that $||f+g||_{\infty} = ||f||_{\infty} + ||g||_{\infty}$ iff either one of $f$ and $g$ is $0$ a.e., or $A_f \cap A_g \neq \varnothing$ and for some $\{x_n\}_{n \in \mathbb{N}} \subset A_f \cap A_g$, there exists $\lambda>0$ such that $b=\lambda \, a$, where $f(x_n) \to a$ and $g(x_n) \to b$.

Here, we set $A_f=\{\{x_n\}_{n \in \mathbb{N}}\}$ where $\{x_n\}_{n \in \mathbb{N}} \subset X$ such that $|f(x_n)| \to ||{f}||_{\infty}$.

I have proved the if part, but still doubtful about the only if part.

Aryaman Maithani
  • 16,136
user825894
  • 41

1 Answers1

1

There is specific and a wider definition of the Minkowski's inequalities. For example Minkowskis Inequalities states a rather different definition. This is another definition Minkowski inequality. This reference offers a proof and references to other types of proofs. This is another informative and reliable reference minkowski inequality. The ask for properties of $f$ and $g$.

This page focues on proving on the Inequality for Sums. This is a answer set to comparable or closely related question on this community: prove minkowskis inequality directly in finite dimensions. But this is restricted for $f$ and $g$ to be convex and somewhat polynomial. And this minkowskis-inequality used the Hölder's inequaltiy twice. This one names it homogenization: proving minkowskis inequality with homogenization. And another one using Holder and Minkowski minkowskis and holders inequality confusion making use to the $=$ concept. This can be continued.

Your concept with the sequences and function sequences is for most of the proofs a lit to much effort, but addmitantly this is sensible if used correct. The first equation is on or in and that has to be taken into account in proper proof theory. But for that $A_f$, $A_g$ and X have to be defined as in the courses or referenced correct. For Minkowski $|(_)|→||||_∞$ is inappropriate. $|(_)|_p→_n→_p||||_∞$ and so on.

Steffen Jaeschke
  • 905