$ L^{1}$ norm of a continuous non-zero function

Question

Is it possible to get a non-zero continuous function which has zero $L^{1}$ norm?

score 1 · Answer 1 · 2016-12-02T06:38:29.757

1

No. If, say, $f(x)<0$, then $\lvert f(x)\rvert>\lvert f(x)\rvert/2$ in an open ball $B$, hence $$\lVert f\rVert_1\ge\int_B \lvert f(x)\rvert\,dx\ge\frac{\mathcal L( B)\lvert f(x)\rvert}{2}$$

edited Dec 02 '16 at 06:38

answered Dec 02 '16 at 06:10

$ L^{1}$ norm of a continuous non-zero function

1 Answers1