Editing Hilbert transform (section)

===Complex structure===
Because {{math|1=H<sup>2</sup> = −I}}  ("{{math|I}}" is the [[identity operator]]) on the real [[Banach space]] of ''real''-valued functions in {{nowrap|<math>L^p(\mathbb{R})</math>,}} the Hilbert transform defines a [[linear complex structure]] on this Banach space. In particular, when {{math|1=''p'' = 2}}, the Hilbert transform gives the Hilbert space of real-valued functions in <math>L^2(\mathbb{R})</math> the structure of a ''complex'' Hilbert space.

The (complex) [[eigenstate]]s of the Hilbert transform admit representations as [[holomorphic function]]s in the upper and lower half-planes in the [[Hardy space]] [[H square|{{math|H<sup>2</sup>}}]] by the [[Paley–Wiener theorem]].