Inequality between $L^2$ norms

Question

Let $X \subset \mathbb{R}^n$ and $Y \subset \mathbb{R}^m$ be two nonempty opens. Let $f \in L^2(X \times Y)$ and, for $x\in X$ fixed, $g_x(y)=f(x,y)$.

Is that true that we always have $\lVert g_x \rVert_{L^2(Y)} \leq \lVert f \rVert_{L^2(X \times Y)}$ ? I think it's pretty obvious that yes because $| g_x |^2$ and $| k(x,y) |^2$ are both positive functions but is there any way to show it properly ?

Answer 1

False. For example take $f(x,y)=\chi_{(0,\frac 1 2)} (x) f_2(y)$ where $\|f_2\|=1$. Your inequality fails for $x$ between $0$ and $\frac 12$.