Weakly closed nonlinear forward maps on Hilbert spaces

Let $X$ and $Y$ be real Hilbert spaces, let $D\subseteq X$ be nonempty, and let $F:D\to Y$ be a forward map. The domain $D$ is weakly closed when it is closed in the weak topology $\sigma (X,X^{\ast })$, the coarsest topology making every bounded real-linear functional on $X$ continuous. The graph of $F$ is sequentially weak-to-weak closed when, for every sequence $(x_n)$ in $D$ and every $x\in X$ and $z\in Y$, the conditions $x_n\rightharpoonup x$ and $F(x_n)\rightharpoonup z$ imply $x\in D$ and $F(x)=z$; here weak convergence means convergence under every bounded real-linear functional on the corresponding Hilbert space. We call $F$ a weakly closed forward map when its domain is weakly closed and its graph is sequentially weak-to-weak closed. No linearity, norm continuity, Fréchet differentiability, or convexity is assumed; linear maps are allowed as special cases.

General forward maps on an arbitrary domain

Real Hilbert spaces

The weak topology

Weak sequence convergence

Christian Clason, Regularization of Inverse Problems, Chapter 10

Benedikt Wirth, Inverse Problems, Chapter 10