I will explain basic concepts/problems of complex analysis in infinite dimensions, and survey the few approaches that are available to solve those problems.
@article{JEDP_1998____A8_0, author = {Lempert, L\'aszl\'o}, title = {The Cauchy-Riemann equations in infinite dimensions}, journal = {Journ\'ees \'equations aux d\'eriv\'ees partielles}, year = {1998}, pages = {1-8}, mrnumber = {99k:46081}, zbl = {01808717}, language = {en}, url = {http://dml.mathdoc.fr/item/JEDP_1998____A8_0} }
Lempert, László. The Cauchy-Riemann equations in infinite dimensions. Journées équations aux dérivées partielles, (1998), pp. 1-8. http://gdmltest.u-ga.fr/item/JEDP_1998____A8_0/
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