Abstract
We study the questions of existence and smoothness of demand functions with an infinite number of commodities. The main result obtained, under some hypothesis, is: if a C 1 demand exists in a commodity space B, then B can be given an inner product structure. For example, if B is L p, 1≦ p≦∞, and if there exists a C 1 demand function defined on B then p must be 2. Another result is: if a demand function exists, defined for all prices p and income, then the commodity space must be reflexive. For example, if B is L p and a demand function exists on B, defined for all prices and incomes then 1< p<∞. We also study the cases L ∞ and L 1 with weaker assumptions. We finish the paper proving that the demand function is always defined for a dense set of prices and convenient incomes.
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