What mean $\nabla f_n$ converge weakly to $\nabla f$ in $L^p$?

Question

Let $f\in L^p(\mathbb R^n)$. What mean $\nabla f_n$ converge weakly to $\nabla f$ in $L^p(\mathbb R^n)$ ?

1) Is it that for all $\varphi\in L^{p'}(\mathbb R^n)$ $$\lim_{n\to \infty }\int |\nabla f_n-\nabla f|\varphi=0$$ or

2) for all $\varphi\in L^p(\mathbb R^n,\mathbb R^n)$ $$\lim_{n\to \infty }\int |(\nabla f_n-\nabla f)\cdot \varphi|=0\ \ ?$$

Answer 1

Neither. It's the statement

for all $\varphi \in L^{p'}(\mathbb{R}^n, \mathbb{R}^n)$, $$\lim_{n \to \infty} \int_{\mathbb{R}^n} (\nabla f_n - \nabla f) \cdot \varphi = 0$$

Note the $L^{p'}$ instead of $L^p$, and the lack of absolute values in the integral.