Let $w \in L^p(\Omega)$, $K$ the set of constants. Consider the quotient space $L^p/K$. Endowed with the norm
$\|[w]\|_{p\sim}=\inf{c \in K}\|w+c\|_{p}$ where $[w] \in L^p/K$ is a class and $w$ is a representative of this class. Notice that as $K$ is closed $L^p / K$ is Banach.
I want to know what the error is in the next argument .
$\|w\|_{p} \leq \|w+c\|_{p}+\|c\|_{p}$ Taking the infimum on $c \in K$, and since $\inf A + B=\inf A +\inf B$ if $A$ and $B$ are bounded below, then we get
$$\inf{c\in K}\|w\|_p=\|w\|_{p}\leq \inf{c \in K}\|w+c\|_p+\inf{c \in K}\|c\|_p$$ $$\leq \|[w]\|_{p\sim}$$
but this implies that $\|[w]\|_{p\sim}=\|w\|_p$. Which is absurd.
If anyone knows how to define another norm or where I can read about this topic please leave a comment.