Ytukyg

Consider $L^p(mathbb{R})$ where $0<p<1$.

Why there are no bounded linear functional on $L^p(mathbb{R})$?
i.e.
If $l$ is linear functional $l:L^p(mathbb{R})to mathbb{C}$ such that $|l(f)|leq M|f|_{p}$ for all $f$ and some $M>0$, then $l=0$.
Why?
Hint: Let $F(x)=l(mathcal{X}_{[0,x]})$, and consider $F(x)-F(y)$

This question surely was already answered here. In the case $0 <p <1$ the usual definition of the $L^p$-norm is not a norm, because the $Delta$-inequality isn't satisfied. If $0< p <1$ the space $L^p(mathbb{R})$ is only a metric space with invariant metric given by
$$|f|_p := int |f|^p d lambda.$$
If $l$ is a continuous functional, we know that $|l(f)| < 1$ for $f in B_delta(0)$ and some $delta >0$. By rescaling and using linearity, we see that $$|l(f)| le M |f|_p^{1/p}$$
with $M= delta^{-1}$. Now decompose $[a,b]$ in $n$ disjoint intervals $I_1,ldots,I_n$ with lenght $(b-a)/n$. Then we have
$$|l(1_{[a,b]})| le sum_{k=1}^n |l(1_{I_i})| le nM |l(1_{I_i})|_p = (b-a)^{1/p} n^{1-1/p}.$$
Since $0<p<1$, we see that $1-1/p$ is negative. Thus, letting $n rightarrow infty$, we get $l(1_{[a,b]}) =0$. Since all linear combinations of such intervals are dense in $L^p$ (also for $0<p<1$), we see that $l$ is trivial.