Need an explanation for the dual maps in the book of Linear Algebra by Werner Greub In the book of Linear Algebra by Werner Greub at page 68, there is an example, which is

As an example of dual mappings consider the dual pairs $E^*$, $E$ and $E^* / E_1^\perp$, $E_1$, where $E_1$ is the subspace of E and let $\pi$ be the canonical projection of $E^*$ onto $E^* / E_1^\perp$. $$\pi :E^* \to E^* / E_1^\perp.$$ Then the canonical injection  $$i: E_1 \to E$$  is dual to $\pi$. In fact if $x\in E_1$, and $y^* \in E^*$ are arbitrary, we have  $$\langle y^*, i(x) \rangle = \langle y^*, x \rangle = \langle \bar y^*, x \rangle = \langle \pi y^*, x \rangle,$$ where $y^*$ is the representative of the class $\bar y^*$.

But I didn't get it how $\langle y^*, x \rangle = \langle \bar y^*, x \rangle$, so can someone explain to me this with more elementary concepts ?
 A: So Greub defines duality for a bilinear map $\langle \cdot, \cdot \rangle : E^* \times E \to \Gamma$ where $\Gamma$ is the underlying field and $E, E^*$ are $\Gamma$-vector spaces.
In this context, we let $E_1$ be a subspace of $E$ and let $E_1^\perp$ be the subspace of $E^*$ of all $y^* \in E^*$ such that $\langle y^*, x \rangle = 0$ for all $x \in E_1$. We then define a bilinear function $\langle \cdot, \cdot \rangle : E^*/E_1^\perp \times E_1 \to \Gamma$ by
$$ \langle \bar y^*, x \rangle = \langle y^*, x \rangle$$
for some $y^* \in \bar y^*$. This is well defined because if $y_1^*, y_2^* \in \bar y^*$ are two representatives then (by definition) $y_2^* - y_1^* \in E_1^\perp$ and hence for all $x \in E_1$,
$$ \langle y_1^*, x \rangle = \langle y_1^*, x \rangle + \underbrace{\langle y_2^* - y_1^*, x \rangle}_0 = \langle y_2^*, x \rangle. $$

Example
Let $E = \operatorname{span}\{e_1, e_2\}$ and $E^* = \operatorname{span}\{e_1^*, e_2^*\}$. Define $\langle \cdot, \cdot \rangle : E^* \times E \to \Gamma$ by
$$ \langle ae_1^* + be_2^*, ce_1 + de_2 \rangle = ac + bd. $$
Let $E_1 = \operatorname{span}\{e_1\}$. Then $E_1^\perp = \operatorname{span}\{e_2^*\}$ and we have the bilinear pair $E^*/E_1^\perp \times E_1 \to \Gamma$ given by
$$ \langle a\bar e_1^*, ce_1 \rangle = \langle ae_1^* + be_2^*, ce_1 \rangle = ac $$
where $a\bar e_1^* = \{ae_1^* + be_2^* : b \in \Gamma\}$. Notice that the element of this coset (the value of $b$) does not affect the bilinear map.
