Riez representation theorem does not hold on infinite-dimensional vector spaces example

Question

Show that the Riesz Representation Theorem does not hold on infinite-dimensional vector spaces without any hypotheses on the vector space V and linear functional.

Answer 1

Put on the vector space $V$ of all real sequences $(x_n)_{n\geq1}$ whose entries are almost all zero the evident inner space, and consider the map $V\to\mathbb R$ which maps such a sequence to the sum of its entries.