  In this paper, we study the Fenchel-Rockafellar duality and the Lagrange
duality in the general frame work of vector spaces without topological
structures. We utilize the geometric approach, inspired from its successful
application by B. S. Mordukhovich and his coauthors in variational and convex
analysis (see \cite{CBN21,CBNC,CBNG22,CBNG,m-book,mn-book}). After revisiting
coderivative calculus rules and providing the subdifferential maximum rule in
vector spaces, we establish conjugate calculus rules under qualifying
conditions through the algebraic interior of the function's domains. Then we
develop sufficient conditions which guarantee the Fenchel-Rockafellar strong
duality. Finally, after deriving some necessary and sufficient conditions for
optimal solutions to convex minimization problems, under a Slater condition via
the algebraic interior, we then obtain a sufficient condition for the Lagrange
strong duality.
