## Theory of multipliers in spaces of differentiable functions

Doing this gives,. The upper limit on the right seems a little tricky but remember that the limit of a constant is just the constant. Note that the function is probably not a constant, however as far as the limit is concerned the function can be treated as a constant. We also wrote the numerator as a single rational expression. This step is required to make this proof work.

### Department of Mathematics

Note that all we did was interchange the two denominators. Since we are multiplying the fractions we can do this. This gives,. The final limit in each row may seem a little tricky. Recall that the limit of a constant is just the constant. By definition we have,. So, define,. Nothing fancy here, but the change of letters will be useful down the road.

## Description

Differentiable surjections between Banach spaces. Approximation of continuous functions on Banach spaces by real-analytic functions without critical points.

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Approximation of differentiable functions in Banach spaces, for the fine topology. Extension of differentiable functions between Banach spaces. Fine and uniform approximation of convex functions and smooth surgery of convex bodies in Banach spaces.

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## AMS :: Transactions of the American Mathematical Society

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Notes , v.