Construct a sequence of functions $f_n\in L_1 (\mathbb{R}) \cap L_2 (\mathbb{R})$, $n=1,2,...$ such that $\frac{||f_n||_2}{||f_n||_1} \rightarrow 0$.

Question

I am having some trouble constructing the required sequences! Help!

Construct a sequence of functions $f_n\in L_1 (\mathbb{R}) \cap L_2 (\mathbb{R})$, $n=1,2,...$ such that $\frac{||f_n||_2}{||f_n||_1} \rightarrow 0$ as $n \rightarrow \infty$.

Justify your example. Here $L_1(\mathbb{R})$ and $L_2(\mathbb{R})$ stand for the spaces of Lebesgue measurable functions $f: \mathbb{R} \rightarrow \mathbb{R}$ for which $\int |f|^2 d \lambda$ and $\int |f| d \lambda$ respectively are finite, where $\lambda$ is the Lebesgue measure on $\mathbb{R}$.

Answer 1

Let $f_n(x) = \frac{1}{n}\chi_{[0,n]}(x).$ Then $\|f_n\|_1 = 1$ and $\|f_n\|_2 = \frac{1}{\sqrt n}.$