dilation operator on $L^2({\mathbb{R}})$ is continuous

Question

Prove the statement let $D:\mathbb{R}^+\rightarrow L^2(\mathbb{R})$ defined by $D(a)=f_a$ and $f_a(x)=\frac{1}{\sqrt{a}}f(\frac{x}{a})$, where $f\in L^2(\mathbb{R})$ then the mapping $D$ is continuous on $\mathbb{R}^+.$

Answer 1

Yes. You can approximate $f$ by a continuous function $g$ whose support is a compact subset of $(0,\infty)$ and it is fairly starightforward to verify continuity of $a \to g_a$. (The given function becomes a uniform limit of continuous functions).