Approximation of differentiable functions on a Hilbert space. III. (English) Zbl 0627.46044

We complete the proof of the theorem which was announced earlier: if \(\Omega\) is an open set in a separable real Hilbert space \({\mathfrak H}\), F is a real Banach space, \(f\in C^ k(\Omega,F)\) where differentiability is understood in the Fréchet sense, k being an integer \(\geq 0\), and \(\epsilon\) (\(\cdot)\) is a positive continuous function on \(\Omega\) ; then there is a function \(g\in C^{\infty}(\Omega,F)\) such that for all integers \(j\in [0,k]\), \(\| D^ jg(x)-D^ jf(x)\| <\epsilon (x)\forall x\in \Omega.\)
With this proof we thus establish: \(C^{\infty}(\Omega,F)\) is dense in \(C^ k(\Omega,F)\) in the \(C^ k\)-fine topology on \(C^ k(\Omega,F)\). [For part II see Contemp. Math. 54, 17-33 (1986; Zbl 0607.46019).]


46E40 Spaces of vector- and operator-valued functions
41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)
46E10 Topological linear spaces of continuous, differentiable or analytic functions


Zbl 0607.46019
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