For $p>1,$ the sequence space $\ell_p$ is defined as $$\ell_p=\left\{ x=(x_i)_{i\in\mathbb{N}} : \sum_{i=1}^\infty |x_i|^p < \infty \right\}.$$
A classical duality theorem of $\ell_p$ space is that $\ell_p^* \cong \ell_q$ where $1/p + 1/q = 1$ and $\cong$ means isometrically isomorphic and $\ell_p^*$ means dual space of $\ell_p.$
One way to show the isometrically isomorphism is the following:
For any $\eta=(\eta_i)_{i\in\mathbb{N}} \in \ell_q,$ define a map $\phi_{\eta}:\ell_q\to\mathbb{R}$ given by the dual space action, that is, $$\phi_\eta(\xi) = \sum_{i=1}^\infty \eta_i \, \xi_i$$ where $\xi=(\xi_i)_{i\in\mathbb{N}} \in \ell_q.$
Then one proceeds to show that $\|\eta\|_q = \| \phi_\eta \|$ to obtain into isometry from $\ell_q$ into $\ell_p*.$
Question: What is the intuition behind the dual space action? When I try to show that $\ell_q \cong \ell_p^*$ myself, I could not think of the action. So I am wondering what triggers one to think of the dual space action.
One way that someone may be led to this mapping $\ell_q\to(\ell_p)^*$ is via Hölder's inequality. If $x=(x_i)\in\ell_p$ and $y=(y_i)\in\ell_q$ for $1<p,q<\infty$ and $p^{-1}+q^{-1}=1$, then Hölder's inequality yields $$\left|\sum_{i=1}^\infty x_iy_i\right|\leq\sum_{i=1}^\infty |x_iy_i|\leq\|x\|_p\|y\|_q.$$ Hence the map $\varphi_y:\ell_p\to\mathbb R$ given by $\varphi_y(x)=\sum_{i=1}^\infty x_iy_i$ is in $(\ell_p)^*$. The natural question to ask at this point is whether or not there are any other elements of $(\ell_p)^*$. Hence the result you are asking about.