  We consider semi-infinite linear programs with countably many constraints
indexed by the natural numbers. When the constraint space is the vector space
of all real valued sequences, we show the finite support (Haar) dual is
equivalent to the algebraic Lagrangian dual of the linear program. This settles
a question left open by Anderson and Nash [Linear programming in infinite
dimensional spaces : theory and applications, Wiley 1987]. This result implies
that if there is a duality gap between the primal linear program and its finite
support dual, then this duality gap cannot be closed by considering the larger
space of dual variables that define the algebraic Lagrangian dual. However, if
the constraint space corresponds to certain subspaces of all real-valued
sequences, there may be a strictly positive duality gap with the finite support
dual, but a zero duality gap with the algebraic Lagrangian dual.
