Definition. Let $X$ be a finite vector space with inner product $\langle\cdot\vert\cdot\rangle_{X}$, and consider the extended real function $F: X\rightarrow \mathbb R_{\infty} := \mathbb R\cup \{\pm \infty\}$. Then $F$ is lower semi-continuous if for any sequence $\left\{ x_n \right\}_n \subset X$ with $x_n\rightarrow x$, it holds: $$F(x) \leq \lim\inf_{n\to\infty} F(x_n).$$
Now, let's say I have a function of two variables, i.e. $\Phi: X \times Y\rightarrow \mathbb R$. How then would the condition for lower semicontinuity look like? Would we say that $\Phi$ is lower semi-cont. if for any sequence $\left\{ \left( x_n, y_n \right) \right\} \subset X \times Y$ with $\left( x_n, y_n\right)\rightarrow \left( x, y\right)$, it holds $\Phi(x, y) \leq \lim\inf_{n\to\infty}\Phi(x_n, y_n)$?
Edit. I would like to understand how to approach showing that the following function in two variables is (weakly) lower semicontinuous? Let $X$, $Y$ be finite Hilbert spaces, $F: X\rightarrow \mathbb R_{\infty}$ convex, $A: X\rightarrow Y$ a linear and bounded mapping and $b\in Y$. Now we consider $\Phi(x, p) := \begin{cases} F(x) \quad \text{if} \ Ax = b-p \\ \infty \quad \text{otherwise}\end{cases} $.
In order to check (weakly) lsc, I can choose a sequence $\left\{ \left( x_n, p_n\right) \right\}$, assuming that $(x_n, p_n) \rightharpoonup (\tilde x, \tilde p)$, i.e.
$$\langle (x_n, p_n), (\bar{x}, \bar{p})\rangle \rightarrow \langle (\tilde{x}, \tilde{p}), (\bar{x}, \bar{p})\rangle \quad \forall (\bar{x}, \bar{p}) \in X\times Y,$$
and now I have to show
$$\Phi(\tilde{x}, \tilde{p}) = \begin{cases} F(\tilde{x}) \quad \text{if} \ A\tilde{x} = b-\tilde{p} \\ \infty \qquad \text{otherwise}\end{cases} \leq \\ \lim\inf_{n\to\infty} \Phi(x_n, p_n) = \lim\inf_{n\to\infty}\left\{ \begin{cases} F(x_n) \quad\text{if} \ Ax_n = b-p_n \\ \infty, \qquad \text{otherwise} \end{cases} \right\}$$
Unfortunately, I am not quite sure I see why this inequality has to hold.