properties of these functions on full open sets. Examples If f ∈ U, then f has Gateaux derivatives of all orders, since for x ∈ U and h1, ..., hk ∈ X, the k-th order Gateaux derivative Dkf(x){h1, ..., hk} involves only iterated directional derivatives in the span of the hi, which is a finite-dimensional space. In this case, the iterated Gateaux derivatives are multilinear in the hi, but will in general fail to be continuous when regarded over the whole space X. Furthermore, a version of Taylor's theorem holds: Here, is the homogeneous polynomial of degree n in y associated with the multilinear operator Dnf(x). The convergence of this series is not uniform. More precisely, if V ⊂ X is a fixed finite-dimensional subspace, then the series converges uniformly on sufficiently small compact neighborhoods of 0 ∈ Y. However, if the subspace V is permitted to vary, then the convergence fails: it will in general fail to be uniform with respect to this variation. Note that this is in sharp contrast with the finite dimensional case. Hartog's theorem holds for Gateaux holomorphic functions in the following sense: If f : (U ⊂ X1) × (V ⊂ X2) → Y is a function which is separately Gateaux holomorphic in each of its arguments, then f is Gateaux holomorphic on the product space. Hypoanalyticity A function f : (U ⊂ X) → Y is hypoanalytic if f ∈ HG(U,Y) and in addition f is continuous on relatively compact subsets of U. Holomorphy A function f ∈ HG(U,Y) is holomorphic if, for every x ∈ U, the Taylor series expansion (which is already guaranteed to exist by Gateaux holomorphy) converges and is continuous for y in a neighborhood of 0 ∈ X. Thus holomorphy combines the notion of weak holomorphy with the convergence of the power series expansion. The collection of holomorphic functions is denoted by H(U,Y). Locally bounded holomorphy A function f : (U ⊂ X) → Y is said to be locally bounded if each point of U has a neighborhood whose image under f is bounded in Y. If, in addition, f is Gateaux holomorphic on U, then f is locally bounded holomorphic. In this case, we write f ∈ HLB(U,Y). References Richard V. Kadison, John R. Ringrose, Fundamentals of the Theory of Operator Algebras, Vol. 1: Elementary theory. American Mathematical Society, 1997. . (See Sect. 3.3.) Soo Bong