It is a well-known fact that, if V is a vector space over a field $k$, then $V^\star \otimes V^\star$ embeds into $(V \otimes V)^\star$.

It turns out to be an isomorphism when $V$ is a finite dimensional vector space.

Why is this finite dimensionnal restriction is necessary?

More explicitely, where is the mistake in the following reasonning?

Let $\mathcal{B}$ be (an infinite) basis of $V$. For all $b_0 \in \mathcal{B}$, we consider the map $b_0^\star : V \longrightarrow k$ defined for all $b \in \mathcal{B}$ by $$ b_0^\star(b) = \left \{ \begin{array}{ll} 1 & \text{, if }b = b_0\\ 0 & \text{, otherwise} \end{array} \right. $$ Consequently, $\mathcal{B}^\star = \{b^\star, b \in \mathcal{B}\}$ is a basis of $V^\star$.

Simillarly, we also consider for all $b_{01}, b_{02} \in \mathcal{B}$ the map $(b_{01} \otimes b_{02})^\star : V \otimes V \longrightarrow k$ defined for all $b_1, b_2 \in \mathcal{B}$ by $$ (b_{01} \otimes b_{02})^\star(b_1 \otimes b_2) = \left \{ \begin{array}{ll} 1 & \text{, if }b_{01} = b_1 \text{ and } b_{02} = b_2\\ 0 & \text{, otherwise} \end{array} \right. $$ Consequently, $\mathcal{C}^\star = \{(b_1 \otimes b_2)^\star, b_1, b_2 \in \mathcal{B}\}$ is a basis of $(V \otimes V)^\star$.

Consequentlu, the map $\varphi: V^\star \otimes V^\star \longmapsto (V \otimes V)^\star$ defined by $\varphi(b_1^\star \otimes b_2^\star) = (b_1 \otimes b_2)^\star$ for all $b_1, b_2 \in \mathcal{B}$ is well-defined and bijective, with an inverse bijection defined by $\varphi^{-1}\big ( (b_1 \otimes b_2)^\star \big ) = b_1^\star \otimes b_2^\star$.

Finaly, do we know an explicit example showing that $V^\star \otimes V^\star$ is not isomorphic to $(V \otimes V)^\star$ in infinite dimension?