Let $(\Omega,\mathcal A,\mu)$ be a finite measure space, $1\le p,q\le\infty$ with $\frac1p+\frac1q=1$, $$\mathfrak m(f,g):=\int fg\:{\rm d}\mu\;\;\;\text{for }(f,g)\in L^p(\mu)\times L^q(\mu)$$ and $$T_{p,\:q}:L^p(\mu)\to L^q(\mu)'\;,\;\;\;f\mapsto\mathfrak m(f,\;\cdot\;).$$
Question 1: How can we show that $T_{p,\:q}$ is an isometry?
Question 2: What is the relation between the adjoint $T_{p,\:q}:L^q(\mu)''\to L^p(\mu)'$ and $T_{q,\:p}$?
Q1: By Hölder's inequality, $$\left\|T_{p,\:q}f\right\|_{L^q(\mu)'}\le\left\|f\right\|_{L^p(\mu)}\;\;\;\text{for all }f\in L^p(\mu)\tag1.$$ If $p,q<\infty$ and $f\in L^p(\mu)\setminus\{0\}$, $$g:=\frac{\overline f}{|f|}\left(\frac{|f|}{\left\|f\right\|_{L^q(\mu)}}\right)^{\frac qp}\in L^q(\mu)\tag2$$ with $\left\|g\right\|_{L^q(\mu)}=1$ and hence it actually holds equality in $(1)$.
But how do we obtain equality in $(1)$ if $p=\infty$ or $q=\infty$?
Q2: Let $\iota_p$ denote the canonical embedding of $L^p(\mu)$ into $L^p(\mu)''$. Assuming that $T_{q,\:p}$ is an isometry (hence injective) even when $p=\infty$ or $q=\infty$ and using that "the inverse of the adjoint is the adjoint of the inverse", it should hold \begin{equation}\begin{split}\left\langle\varphi,\left(T_{q,\:p}'\right)^{-1}T_{p,\:q}f\right\rangle&=\left\langle T_{q,\:p}^{-1}\varphi,T_{p,\:q}f\right\rangle=\int fT_{q,\:p}^{-1}\varphi\:{\rm d}\mu\\&=\left\langle f,T_{q,\:p}T_{q,\:p}^{-1}\right\rangle=\langle f,\varphi\rangle=\langle\varphi,\iota_pf\rangle\end{split}\tag3\end{equation} for all $(f,\varphi)\in L^p(\mu)\times L^q(\mu)$ and hence $$\iota_p=\left(T_{q,\:p}'\right)^{-1}\circ T_{p,\:q}\tag4.$$
But can we say more?