Diference between dual spaces

Question

What is the diference between Algebraic Dual Space and Topologic Dual Space in Normed Vector spaces with $dim=\infty$

Answer 1

By definition the difference is clear:

The algebraic dual space of a vector space $V$ is the space of all linear functionals on $V$.

The topological dual space is the space of continuous linear functionals on $V$.

The problem is to show that there exists non continuous linear functional so that the topological dual space is a proper subspace of the algebraic dual space.

For finite dimensional vector spaces we can prove that any linear functional is bounded (so it's continuous), but for infinite dimensional vector spaces we can prove, using the axiom of choice, that there exist unbounded linear functionals, so the two dual spaces do not coincide. But the proof is not constructive, in the sense that we cannot exibit explicitly a not continuous linear functional for a complete space.