  In this paper, we study the Cauchy problem of the 2D incompressible
magnetohydrodynamic equations in Lei-Lin space. The global well-posedness of a
strong solution in the Lei-Lin space $\chi^{-1}(\mathbb{R}^2)$ with any initial
data in $\chi^{-1}(\mathbb{R}^2)\cap L^2(\mathbb{R}^2)$ is established.
Furthermore, the uniqueness of the strong solution in $\chi^{-1}(\mathbb{R}^2)$
and the Leray-Hopf weak solution in $L^2(\mathbb{R}^2)$ is proved.
