## 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).]

### MSC:

 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

### Keywords:

C$${}^ k$$-fine topology

Zbl 0607.46019
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### References:

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