91Ó°ÊÓ

The positive cone, denoted as \(C\), is essentially a subset of \(\ell_2(\Gamma)\) where all elements are non-negative. Specifically, \(C = \{x = (x_\gamma) \in \ell_{2}(\Gamma) ; x_\gamma \geq 0 \text{ for all } \gamma \in \Gamma\}\). This cone includes all sequences whose elements are zero or positive. Understanding this concept is crucial, as it helps set up the stage for exploring mappings such as \(P\) and their differentiability properties.

A function is Gâteaux differentiable at a point if we can approximate it using a linear map around that point in a local region. Formally, a function \(f\) is Gâteaux differentiable at \(x \in \ell_2(\Gamma)\) if there exists a continuous linear map \(L\) such that \[ \lim_{t \to 0} \frac{f(x + th) - f(x)}{t} = L(h) \] for every direction \(h \in \ell_2(\Gamma)\). Here, variations are considered around points in the positive cone.
However, due to the nature of the projection \(P\) onto \(C\), differentiability would necessitate this linear map to hold over uncountable dimensions, which is not feasible. Thus, when \(\Gamma \) is uncountable, \(P\) fails to be Gâteaux differentiable anywhere.

Fréchet differentiability is a stronger condition than Gâteaux differentiability. A function is Fréchet differentiable if it is Gâteaux differentiable and the linear map serves as a good approximation in the normed sense.
To put it formally, \(f\) is Fréchet differentiable at \(x \in \ell_2(\Gamma)\) if there exists a bounded linear operator \(A\) such that \[ \lim_{\|h\| \to 0} \frac{\|f(x + h) - f(x) - A(h)\|}{\|h\|} = 0 \] Due to the infinite dimensionality of \(\Gamma\), \(P\)'s nature inhibits this approximation with any consistent linear operator, leading to its nowhere Fréchet differentiability in infinite dimensional contexts.

A map \(P\) is considered Lipschitz if there exists a constant \(L\) such that for all \(x, y \in \ell_2(\Gamma)\), the distance between \(P(x)\) and \(P(y)\) is proportionally bounded by the distance between \(x\) and \(y\). Mathematically, this is expressed as \[ \|P(x) - P(y)\| \leq L \|x - y\| \] This property ensures the function \(P\) does not drastically distort distances and behaves in a regular and controlled manner.
In our context, the projection map \(P\) can be seen assigning each point in \(\ell_2(\Gamma)\) to its nearest point in the positive cone \(C\). For any component \(x_\gamma\), \(P\) sets \(P(x)_\gamma = \max\{x_\gamma, 0\}\).