User:S.anzengruber/Sandbox/Regularization Theory/LSC of Discrepancy Functional

Theorem
Let $(X, \tau_X)$ and $(Y, \tau_Y)$ be Hausdorff spaces.

Let $F: X \to Y$ be a sequentially continuous mapping.

Let $d : Y \times Y \to \R \cup \left\{ {+ \infty} \right\}$ be an extended real-valued functional.

If $d$ is sequentially lower semicontinuous with respect to the product space topology on $Y \times Y$, then:


 * $\displaystyle \mathcal S \left( {x, y} \right) := d \left( {F \left( {x} \right), y} \right)$

is sequentially lower semicontinuous with respect to the product space topology on $X \times Y$.

Proof
Let $\left\langle \left( x_n, y_n \right) \right\rangle_{n \in \N}$ be a sequence in $X \times Y$ such that:


 * $\displaystyle \left( x_n, y_n \right) \to \left( \bar x, \bar y \right)$

This implies that $x_n \to \bar x$ and $y_n \to \bar y$ by Sequence on Finite Product Space Converges to Point iff Projections Converge to Projections of Point.

By the sequential continuity of $F$ it follows that $F \left( {x_n} \right) \to F \left( {\bar x} \right)$.

Thus $\left( {F \left( {x_n} \right), y_n} \right) \to \left( {F \left( {\bar x} \right), \bar y} \right)$ because Sequence on Finite Product Space Converges to Point iff Projections Converge to Projections of Point

By the sequential lower semicontinuity of $d$, we obtain


 * $\displaystyle \mathcal S \left( {\bar x, \bar y} \right) = d \left( {F \left( {\bar x} \right), \bar y} \right) \leq \liminf_{n \to \infty} d \left( {F \left( {x_n} \right), y_n} \right) = \liminf_{n \to \infty} \mathcal S \left( {x_n, y_n} \right)$

Hence the result.

Open question
Can the proof be generalized to hold for sequentially closed mappings $F$?