Let H be a real Hilbert space and $\Phi :H\to H$ be a $C^1$ operator with Lipschitzian derivative and closed range. We prove that $\Phi ^{-1}(0)\neq \emptyset $ if and only if, for each $\epsilon>0$, there exist a convex set $X\subset H$ and a convex function $\psi :X\to \mathbf {R}$ such that $\sup _{x\in X}(\|x\|^2+\psi (x))-\inf _{x\in X}(\|x\|^2+\psi (x))<\epsilon $ and $0\in \overline {{\mathrm {conv}}}(\Phi (X))$.

We prove that for every function $f\,:\,X\,\to \,Y$ , where $X$ is a separable Banach space and $Y$ is a Banach space with RNP, there exists a set $A\,\in \,\overset{\sim }{\mathop{\mathcal{A}}}\,$ such that $f$ is Gâteaux differentiable at all $x\,\in \,S\left( f \right)\backslash A$, where $S\left( f \right)$ is the set of points where $f$ is pointwise-Lipschitz. This improves a result of Bongiorno. As a corollary, we obtain that every $K$-monotone function on a separable Banach space is Hadamard differentiable outside of a set belonging to $\tilde{C}\,;$ this improves a result due to Borwein and Wang. Another corollary is that if $X$ is Asplund, $f\,:\,X\,\to \,\mathbb{R}$ cone monotone, $g\,:\,X\,\to \,\mathbb{R}$ continuous convex, then there exists a point in $X$, where $f$ is Hadamard differentiable and $g$ is Fréchet differentiable.

We study the range of the gradients of a ${{C}^{1,\alpha }}$ -smooth bump function defined on a Banach space. We find that this set must satisfy two geometrical conditions: It can not be too flat and it satisfies a strong compactness condition with respect to an appropriate distance. These notions are defined precisely below. With these results we illustrate the differences with the case of ${{C}^{1}}$-smooth bump functions. Finally, we give a sufficient condition on a subset of ${{X}^{*}}$ so that it is the set of the gradients of a ${{C}^{1,1}}$-smooth bump function. In particular, if $X$ is an infinite dimensional Banach space with a ${{C}^{1,1}}$-smooth bump function, then any convex open bounded subset of ${{X}^{*}}$ containing 0 is the set of the gradients of a ${{C}^{1,1}}$-smooth bump function.

We describe new construction principles for infinite-dimensional Lie groups. In particular, given any measure space $\left( X,\,\sum ,\,\mu \right)$ and (possibly infinite-dimensional) Lie group $G$, we construct a Lie group ${{L}^{\infty }}\left( X,G \right)$, which is a Fréchet-Lie group if $G$ is so. We also show that the weak direct product $\prod{_{i\in I}^{*}{{G}_{i}}}$ of an arbitrary family ${{\left( {{G}_{i}} \right)}_{i\in I}}$ of Lie groups can be made a Lie group, modelled on the locally convex direct sum ${{\oplus }_{i\in I}}L\left( {{G}_{i}} \right)$ .