On Hilbert spaces, the universal Galerkin property is equivalent to essential coercivity plus uniqueness, and on Banach spaces it is tied to the existence of a finite-dimensional Schauder decomposition.
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Galerkin approximation of linear problems in Banach and Hilbert spaces
On Hilbert spaces, the universal Galerkin property is equivalent to essential coercivity plus uniqueness, and on Banach spaces it is tied to the existence of a finite-dimensional Schauder decomposition.