Prove that sum of multiple variable convex function is also convex

Question

Suppose I have a multiple variable function $f(X,Y) = \sum_i \sum_j k_{ij} g(x_i,y_j) + \sum_j h(y_j)$ where $X = \{x_i\}$ and $Y = \{y_j\}$. $k_{ij}$ is a non-negative constant. I want to show whether $f(X,Y)$ is convex. It is difficult to directly show the convexity of $f(X,Y)$, so I attempt to prove that the components $g(x_i,y_j)$ and $h(y_j)$ are both convex in their domains. Does it imply $f(X_,Y)$ is also convex since $f(X,Y)$ is a linear combination of $g(x_i,y_j)$ and $h(y_j)$?