Complexifications of real spaces: general aspects (Q2897180)

Let \(X\) be a linear space. A mapping \(\text{In}:X \to X\) is called an involution if for any \(x_1, x_2 \in X\) and any complex scalar \(c\) we have \(\text{In}(x_1 + x_2) = \text{In}(x_1) + \text{In}(x_2)\), \(\text{In}(cx_1) = \bar{c}\text{In}(x_1)\), and \(\text{In} \circ \text{In}(x_1) = x_1\).NEWLINENEWLINEIn the first part of this paper the authors study a linear space in the four following situations: (i) the space is real and admits a multiplication by complex scalars which does not change the set itself, (ii) the space is real and can be embedded into a wider set which has a multiplication by complex scalars, (iii) the space is complex and admits an involution and thus can be decomposed into ``real'' and ``imaginary'' elements (Proposition 2.6), (iv) a combination of the situations (i)--(iii).NEWLINENEWLINEThe authors also study the situations (i)--(iv) when the linear space in addition, and respectively, has a topology, a norm, and an inner product.NEWLINENEWLINE The final part of the paper is devoted to the study of linear operators (respectively continuous linear operators) between (respectively topological, normed, and inner product) spaces of the type (i)--(iv).