Counterxample of representation theorem

Question

Give an example of non-degenerate scalar product $\langle,\rangle$ on a vector space $V$ such that the application $$\varphi : V\to V^* : v\mapsto L_v$$defined by $L_v(w)=\langle v,w\rangle$ is not an isomorphism.

Answer 1

Consider the vector space $V$ of all infinite sequences $(x_i)_{i\geq0}$ of real numbers with finitely many non-zero terms, and endow it with the inner product such that $$\langle(x_i)_{i\geq0},(y_i)_{i\geq0}\rangle=\sum_{i\geq0}x_iy_i.$$ There is a linear function $\phi:V\to\mathbb R$ such that $$\phi((x_i)_{i\geq0})=\sum_{i\geq0}x_i$$ for all sequences $(x_i)_{i\geq0}\in V$, and you can easily check that it is not in the image of the canonical map $V\to V^*$. Indeed, no vector $(\xi_i)_{i\geq0}$ in $V$ exists such that $$\sum_{i\geq0}x_i\xi_i=\sum_{i\geq0}x_i$$ for all $(x_i)_{i\geq0}\in V$.