Background on Functional Analysis

Let \(\mathcal X\) , \(\mathcal {Y}\) , and \(\mathcal {Z}\) be three sets. In general, a map defined on \(\mathcal X\times \mathcal {Y}\) with values in \(\mathcal {Z}\) is denominated by a letter, let us say, f. The image of any ordered pair \((x, y)\) of \(\mathcal X\times \mathcal {Y}\) is denoted by \(f(x, y)\) . However, in some situations, this image is designated by \(\langle x , y \rangle \) or by \((x, y)\) . For the basic maps on vector spaces, the image of \((x, y)\) is \(x+y\) or xy. These maps, which we will call operations, are denoted by \(+\) and . (i.e. the dot symbol). Let \(\mathbb {K}\) be equal to \(\mathbb {R}\) or \(\mathbb {C}\) .