I am learning about duality in convex optimisation. The Lagrangian is defined as $$L(x, \lambda, \nu) = f_0(x) + \sum_{i=1}^m\lambda_if_i(x) + \sum_{i=1}^p\nu_ih_i(x)$$ where suppose the optimisation problem has $m$ inequality and $p$ equality constraints. The Lagrangian dual function is defined as $$g(\lambda, \nu) = inf_{x \in D}L(x,\lambda,\nu)$$
Given a set of numbers (1-D case), I know that the infimum is the greatest lower bound on this set of numbers. How does the definition of infimum extend to the multivariate case as in the Lagrangian dual function?