Is every integrable function on the real line with compact support also square integrable?

Question

I wonder that whether every integrable function on the real line with compact support is also square integrable ? In other words, is $L^1_c(\mathbb R)\subseteq L^2(\mathbb R)$ holds true? Thanks in advance for any hints.

If $f\in L^2[a,b]$ for some $a,b\in\mathbb R$, then $f\in L^1[a,b]$. ${}\qquad{}$ — Michael Hardy, Apr 27 '14 at 03:04

score 8 · Answer 1 · answered Apr 27 '14 at 03:04

8

Consider $x \mapsto 1/ \sqrt x$

answered Apr 27 '14 at 03:04

Pedro

122,002

or any function $x^a$ for $-1\lt a\le-\frac12$. (+1) – robjohn Apr 27 '14 at 09:39
2

How does this function have compact support? – user2357112 Apr 27 '14 at 11:02
@user2357112 I left the details to the OP. – Pedro Apr 27 '14 at 12:45

Is every integrable function on the real line with compact support also square integrable?

1 Answers1