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Near-field antennas have been successfully adopted in several wireless applications. To exploit the high reconfigurability of array antennas, multiple synthesis techniques for arrays operating in the near-field region have been proposed. Building upon previous works on eigenmode expansions of the radiated fields, two synthesis methods for the excitations of near-field arrays based on the definition of an inner product on the electromagnetic fields are investigated: the "maximum norm" and "minimum error field norm" methods. The "maximum norm" method computes the array excitations that maximize either the active power flow through a target surface or the electric/magnetic energy stored in an assigned volume, depending on the adopted inner product. The performance of the maximum active power flow method is compared with the one of the simpler conjugate phase method. Furthermore, the limit solution achieved when the target surface reaches the far-field region is compared against the "maximum Beam Collection Efficiency" method. The "minimum error field norm" method allows to synthesize a given target field. As an example, the latter method is used to find the optimal excitation of a Plane Wave Generator with a spherical quiet zone. The effectiveness and performance of the discussed synthesis methods are validated through numerical simulations.