Exercice: let $(\Omega,\mathcal{A},\mu)$ be a measurable space, $p\in[1,+\infty[$ and let $g\in L^q(\Omega)$ with $q$ is the conjugate exponent of $p,$ let $T$ such that $T:L^p(\Omega)\to\mathbb{C}$ defined by $T(f)=\int_\Omega f\overline{g}\,d\mu$, I want to prove that $T$ is well defined and continious such that we have :$\|T\|\leq \|g\|_q$ ?
My Attempt: I have tried to reformualte the Holder inequality like this :
\begin{eqnarray*} \int_{\Omega}f\bar{g} \,d\mu & \leq & \left(\int_\Omega f^p \, d\mu\right)^{1/p} \left(\int_\Omega \bar{g}^q \, d\mu\right)^{1/q}\\ &\leq&\|f\|_p\|\bar{g}\|_q \end{eqnarray*} Now $\|f\|_p\|\bar{g}\|_q$ is finite implies that $T$ is defined , for continuity I have used the fact that $T$ is measurable implies that is continious , But my problem how I can show that $\|T\|\leq \|g\|_q$ , I have used the assumption $1\leq p <q < \infty$ implies $\ell^p\subset \ell^q$ which prove the result , But am not care about this way?
Your question is not realted to $\ell^{p}$ spaces (which are different from $L^{p}$ spaces). You don't need $p<q$ for this.
You already know (by Holder's inequality) that $|Tf|\leq \|f\|_p \|g\|_q$. [ Note that $|\overline {g}|=|g|$ so you can drop the conjugate].
A linear map $T$ from one normed linear space $X$ to another normed linear space $Y$ is continuous iff it is bounded in the sense $\|Tx\| \leq C\|x\|$ for some finite constant $C$. By definition $\|T\|$ is the infiumum of all such constants $C$. Hence, taking $C=\|g\|_q$, $T$ is bounded in our case and $\|T\| \leq \|g\|_q$.