= Solution
Fix paired orthonormal bases in the two $d$-dimensional <Hilbert spaces>. A <maximally entangled state> is
$$
\boxed{|\Psi_{AB}\rangle=\frac1{\sqrt d}\sum_{j=1}^d|j\rangle_A\otimes|j\rangle_B.}
$$
It is normalized, and the <partial traces> of its rank-one <density operator> are $\rho_A=\rho_B=I/d$. Thus all its <Schmidt coefficients> are $1/\sqrt d$ and its <entanglement entropy> is $\log_2d$. No pure bipartite state of these dimensions can have larger entanglement entropy: <nonnegativity of quantum relative entropy> applied to a reduced state and $I/d$ gives $0\leq D(\rho_A\Vert I/d)=\log_2d-S(\rho_A)$. Hence it is maximally entangled.
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