The Fubini-Study map is quantitatively injective: the Hilbert-Schmidt distance between two metrics is bounded by k^n times the W^{2,2} norm of the associated function.
On The Image Of The Hilbert Map
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We talk about the image of the Hilbert map. We show the necessary and sufficient condition that the Hilbert map is surjective.
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Quantitative injectivity of the Fubini--Study map
The Fubini-Study map is quantitatively injective: the Hilbert-Schmidt distance between two metrics is bounded by k^n times the W^{2,2} norm of the associated function.