TITLE: Difference between the algebraic and topological dual of a topological vector space?
QUESTION [12 upvotes]: What is the difference between the algebraic and the topological dual of a topological vector space, such as for example the Euclidean space $\mathbb{H}$?
I am interested in intuitive as well as in detailled technical answers.

REPLY [9 votes]: In finite dimensions, there is no difference; see the post Let $X$ be a finite dimensional normed space. Does the algebraic dual space $X^*$ and the dual space $X'$ coincide?
In infinite dimensions, the difference is huge. For example, the topological dual (the space of all continuous linear functionals) of a Hilbert space is the Hilbert space itself, by the Riesz representation theorem, while the algebraic dual (the space of all linear functionals) is vastly bigger since there are lots of non-continuous linear functionals. 
For example, one can use Zorn's lemma to extend an orthonormal basis (in the Hilbert space sense) to an algebraic basis by adding inductively elements that cannot be written as finite linear combinations of basis elements already chosen. Then one can map each basis element to an arbitrary number and extend by linearity to a linear functional. Most of these are not continuous since the continuous ones are already fully determined by the image of the orthonormal basis. 
This only works in infinite dimensions since an orthonormal basis is not already an algebraic basis iff the dimension is infinite.