I was reading a paper and stumbled across the terms $$f^\varepsilon\to f \text{ in } L^{p}_{loc} \Bigl( [0,+\infty) , \left[ L^p_{loc}(\mathbb{R}^2) \right]^* \Bigr)$$ and $$f^\varepsilon\rightharpoonup^* f \text{ in } L^{\infty} \Bigl( [0,+\infty) , W^{s,q}_{loc}(\mathbb{R}^2) \Bigr)$$
1st question; How is $\left[ L^p_{loc}(\mathbb{R}^2) \right]^*$ defined, and is it equal to $L^q_{loc}(\mathbb{R}^2)$ where $\frac{1}{q}+\frac{1}{p}=1$?
2nd question; What do the convergences mean? I am interested only in the absolute basics, only the very definition/meaning.
3rd question; Have you ever came across any of the aforementioned in any book, papers, etc that state their definition and might be helpful to the understanding of their meaning/definition?
paper: I have no link to the particular paper, but I have the links for the paper and the book referenced book and paper