On the Example of Vector-Valued Integrals Let $p>1$. I am looking for a sequence $(f_{n}(x))$ such that
\begin{align*}
\left(\sum_{n}|f_{n}(x)|^{2}\right)^{1/2}&\notin L^{p}(\mathbb{R}^{N}),\\
\sum_{n}a_{n}f_{n}(x)&\in L^{p}(\mathbb{R}^{N}),~~~~\text{for any}~(a_{n})\in l^{2}~\text{with}~\|(a_{n})\|_{l^{2}}\leq 1
\end{align*}
so that
\begin{align*}
\sup_{\|(a_{n})\|_{l^{2}}\leq 1}\left\|\sum_{n}a_{n}f_{n}(x)\right\|_{L^{p}(\mathbb{R}^{N})}<\infty.
\end{align*}
The tricky thing is that, the space $l^{2}$ is self-dual, but when we treat it as the inner one but taking $L^{p}$ integral as outer one, the situation is different.
I also think about the Khintchine inequality, but this does not help.
Any suggestion?
 A: Assuming that $1<p<2$:
Let $(S_{n})$ be a sequence of disjoint sets in $\mathbb{R}^{N}$ such that the volume $|S_{n}|=1$.
Let $f_{n}=(1/n^{1/p})\chi_{S_{n}}$, we compute that
\begin{align*}
\int_{\mathbb{R}^{N}}\left(\sum_{n}|f_{n}(x)|^{2}\right)^{p/2}dx&=\sum_{n}\int_{S_{n}}f_{n}(x)^{p}dx\\
&=\sum_{n}\dfrac{1}{n}\\
&=\infty.
\end{align*}
On the other hand, for any $(a_{n})\in l^{2}$ such that $\|(a_{n})\|_{l^{2}}\leq 1$, we compute that
\begin{align*}
\int_{\mathbb{R}^{N}}\left|\sum_{n}a_{n}f_{n}(x)\right|^{p}dx&=\sum_{n}\int_{S_{n}}|a_{n}|^{p}\cdot\dfrac{1}{n}dx\\
&=\sum_{n}|a_{n}|^{p}\cdot\dfrac{1}{n}\\
&\leq\left(\sum_{n}|a_{n}|^{2}\right)^{p/2}\left(\sum_{n}\dfrac{1}{n^{r}}\right)^{1/r}\\
&\leq\left(\sum_{n}\dfrac{1}{n^{r}}\right)^{1/r},
\end{align*}
where $1/(2/p)+1/r=1$.
Now assuming that $p\geq 2$:
Note that $\|(a_{n})\|_{l^{2}}\leq 1$ implies that $|a_{n}|\leq 1$ for each $n=1,2,...$ and hence
\begin{align*}
\int_{\mathbb{R}^{N}}\left|\sum_{n}a_{n}f_{n}(x)\right|^{p}dx&=\sum_{n}|a_{n}|^{p}\cdot\dfrac{1}{n}\\
&\leq\sum_{n}|a_{n}|^{2}\\
&\leq 1.
\end{align*}
