An example in $L^p$ space.

Question

I am trying to find two sequence $f_n$ and $g_n$ that converge in $L^2$ to $f$ and $g$ $$f_n \rightarrow f~~ in ~~L^2$$ and $$g_n \rightarrow g~~ in ~~L^2$$

that $f_ng_n \in L^2~~~and~~ fg\in L^2$. but $$f_ng_n$$ not converge to $fg$ in $L^2$. and I don't know when we say $f_n\rightarrow f$in $Lp$ means $||f_n-f||_{Lp}\rightarrow 0$ or $||f_n||_{Lp}\rightarrow ||f||_{Lp}$

Answer 1

Hint: You can look for a sequence $f_n = \lambda_n \mathbf{1}_{A_n}$, where $\lambda_n \in \mathbb{R}$ and $A_n$ is measurable such that:

• $f_n \to 0$ in $L^2$ (that is ${\lambda_n}^2 \,\mu(A_n) \to 0$)
• but $f_n^2 \not\to 0$ in $L^2$ (for instance ${\lambda_n}^4 \,\mu(A_n) \to 1$)