A perspective on the relevance of com plex num bers and conjugate linearity

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Abstract:

The concept of linearity is of some importance in mathematics. Its counterpart, conjugate linearity, is less well known but has an equally important presence in mathematics. At a fundamental level, for example, the representation of a reflection symmetry in R2 as a mapping of complex numbers is conjugate linear. At a more sophisticated level the inner (or scalar) product in a complex inner product space is conjugate linear in one variable: this is reflected in the Reisz representation theorem for a complex Hilbert space H which establishes a conjugate-linear isomorphism between H and the dual space of bounded linear functionals on H. Despite the utility of complex numbers and conjugate linearity it is surprising to hear of referees of applied mathematics journals claiming that 'it would make no difference if multiplication of complex numbers had never been invented' [1]. This may be because the majority of studies of conjugate linearity are aimed at pure mathematicians (for example, [2]). This article aims to demonstrate the relevance of conjugate linearity in mathematics to the non-expert by simple but effective means.

Document Type: Research Article

DOI: http://dx.doi.org/10.1080/002073999287833

Publication date: July 1, 1999

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