Sentence:  The <WSD>dual space</WSD> of a Banach space X is the vector space of continuous linear functions X \rightarrow\mathbb{R}, which are called functionals. Similar notation is used for duality pairing between the Banach space X and its <WSD>dual space</WSD> X': \left\langle u, v\right\rangle is the result of applying the functional u \in X' to v \in X: \left\langle u, v\right\rangle = u(v) explicitly uses the fact that u is a function X \rightarrow\mathbb{R}. 
Meaning of the word marked with <WSD> tags:  The vector space which comprises the set of continuous linear functionals of a given topological vector space.