Several Complex Variables

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Extremal metrics on blowups (2006, background): “This is accomplished by linearizing on a small neighborhood of p the action of K, which is ensured by the following classical result [6] : Proposition 7.”
Affine Processes on Positive Semidefinite Matrices (2009, unknown): “Therefore, and since S++ d is a set of uniqueness in S++ d + iSd , it follows by Vitali’s theorem ([41], Chapter 1, Proposition 7) that the analytic functions Ln converge uniformly on compact subsets of S ++ d + iSd to an analytic limit thereon.”
On blowing up extremal Kähler manifolds (2010, background): “Choose local normal coordinates z near p such that the group T acts by unitary transformations on the unit ball B1 around p (this is possible by linearizing the action, see Bochner-Martin [5] Theorem 8).”
Analysis of the limiting spectral distribution of large dimensional information-plus-noise type matrices (2007, methodology): “Then by the implicit function theorem (Krantz [4] p.”