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But what about the final spaces that architects themselves will occupy?
Let (0< mle Ale M) and let X, Y be two isometries on (mathcal{H}) whose final spaces are orthogonal to each other.
Bhatia and Davis [4] proved that if (0< mle Ale M) and X and Y are two partial isometries on (mathcal{H}) whose final spaces are orthogonal to each other.
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com