In this note, we give a general form of linear functionals on martingale Hardy spaces $\mathcal H_{\Phi}$, where the Young function $\Phi$ has finite power and its conjugate function $\Psi$ is of the form $\Psi(x)=\Psi_1(x^2)$ with $\Psi_1$ being also a Young function. Moreover, we discuss the relationship between the martingale Hardy space $\mathcal H_{\Phi}$ and the spaces $\mathcal K_{\Psi}$.