open mapping theorem


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open mapping theorem

[′ō·pən ′map·iŋ ‚thir·əm]
(mathematics)
A continuous linear function between Banach spaces which has closed range must be an open map.
References in periodicals archive ?
Then by open mapping theorem for analytic functions, [zeta] [member of] [partial derivative]X.
Open Mapping Theorem. If D is open in [R.sup.n] and f is a nonconstant qr function from D to [R.sup.n], we have that f(D) is open set.
By the Open Mapping Theorem of complex analysis, there are some r > 0 and [zeta] [member of] [B.sub.r] ([[lambda].sub.j]), the open disk of radius r and center [[lambda].sub.j], so that z = f ([zeta]).
He starts with Cauchy-Riemann equations in the introduction, then proceeds to power series, results on holomorphic functions, logarithms, winding numbers, Couchy's theorem, counting zeros and the open mapping theorem, Eulers formula for sin(z), inverses of holomorphic maps, conformal mappings, normal families and the Riemann mapping theorem, harmonic functions, simply connected open sets, Runge's theorem and the Mittag-Leffler theorem, the Weierstrass factorization theorem, Caratheodory's theorem, analytic continuation, orientation, the modular function, and the promised Picard theorems.
But this implies immediately (see the proof of the open mapping theorem) that X is isometric to [S.sup.*].
Therefore, [G.sub.L] is continuous and by the open mapping theorem, [G.sup.-1.sub.L] is also continuous.
Calling on the Open Mapping Theorem we conclude that [[PI].sub.q;p] (X, Y) [not equal to] L (X, Y).