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We establish sharp well-posedness and approximation estimates for variationalsaddle point systems at the continuous level. The main results of this notehave been known to be true only in the finite dimensional case. Known spectralresults from the discrete case are reformulated and proved using a functionalanalysis view, making the proofs in both cases, discrete and continuous, lesstechnical than the known discrete approaches. We focus on analyzing the specialcase when the form $a(\cdot, \cdot)$ is bounded, symmetric, and coercive, andthe mixed form $b(\cdot, \cdot)$ is bounded and satisfies a standard$\inf-\sup$ or LBB condition. We characterize the spectrum of the symmetricoperators that describe the problem at the continuous level. For a particularchoice of the inner product on the product space of $b(\cdot, \cdot)$, we provethat the spectrum of the operator representing the system at continuous levelis $\left \{\frac{1-\sqrt{5}}{2}, 1, \frac{1+\sqrt{5}}{2} \right \}$. Asconsequences of the spectral description, we find the minimal length intervalthat contains the ratio between the norm of the data and the norm of thesolution, and prove explicit approximation estimates that depend only on thecontinuity constant and the continuous and the discrete $\inf-\sup$ conditionconstants.

A note on stability and optimal approximation estimates for symmetric saddle point systems