Question about injection on an unbounded space

Question

I have this space $$C_0((0,+\infty))=\left\lbrace u,u\in C((0,+\infty)),\lim_{t\rightarrow +\infty} u(t)=0\right\rbrace$$ with the norm $$||u||_{\infty}=\sup_{t\geq0}|u(t)|$$

how to prove that $C_0((0,+\infty))\hookrightarrow L^2((0,\infty))$ (continuous injection).

Please

Thank you.

Answer 1

Before asking for continuity, we have to check whether the inclusion holds. It's not the case, because the function defined by $$f(x)=\begin{cases}x&\mbox{ if }x\in (0,1),\\ \frac 1{\sqrt x}&\mbox{ if }x\in (1,+\infty), \end{cases}$$ is continuous on $(0,\infty)$, its limit at infinity is $0$, but $f^2$ is not integrable.