Show that $N(f)= \inf\{\|g\|_p+\|h\|_q;f=g+h, g \in L^p(\mathbb{R}), h \in L^q(\mathbb{R})\}$ is a norm on $L^p(\mathbb{R})+L^q(\mathbb{R}) $

Question

Define $N(f)= \inf\{\|g\|_p+\|h\|_q;f=g+h, g \in L^p(\mathbb{R}), h \in L^q(\mathbb{R})\}$.

I want to prove that $N$ is a norm on $L^p(\mathbb{R})+L^q(\mathbb{R})$ but I can't prove that $N(f)=0 $ implies $f= 0$.

Could this result be proved without a further hypothesis? (Such as "compatibility” assumption...). Could you please prove or give me some references for the proof of this result? I was trying to prove it but there is no progress.

After that, we have $L^p(\mathbb{R})+L^q(\mathbb{R})$ with $N$ is a Banach space since $(L^p(\mathbb{R}), \|.\|_p)$ and $(L^q(\mathbb{R}), \|.\|_q)$ are Banach spaces. Is this correct?

Thank you so much.

Answer 1

You had a very good intuition when you mentioned a compatibility assumption.

A precise result regarding how to norm a sum of vector spaces is given in this article. Namely, the norm is what you say, and the compatibility assumption is that

For all sequences $f_k$ in $L^p+L^q$, if

• $f_k$ converges to $f$ in $L^p$

• $f_k$ converges to $\tilde f$ in $L^q$

then $f=\tilde f$.

Note that in your case it is implicit that "$=$" means "equal almost everywhere". In the same article it is explained that for $L^p$ and $L^q$ this compatibility does occur, since convergence in any $L^r$ implies convergence almost everywhere after passing to a subsequence.

Answer 2

If a sequence converges in $L^{p}$ then there is a subsequence which converges almost everywhere. $N(f)=0$ implies there exist sequences $\{g_n\} \subset L^{p}$,$\{h_n\} \subset L^{q}$ such that $f=g_n+h_n$ , $||g_n||_p \to 0$ and $||h_n||_q\to 0$. First choose a subsequence along which $\{g_n\} \to 0$. Then look at the corresponding subsequence of $\{h_n\}$ and get a subsequence which $\to 0$ a.e.. By doing this we get one sequence $\{n_k\}$ increasing to $\infty$ such that $\{g_{n_k}\} \to 0$ and $\{h_{n_k}\} \to 0$. Since $f=g_{n_k}+g_{n_k}$ for each k we get $f=0$ a.e. which means f is the zero element of $L^{p}+L^{q}$.