= Solution
The <adjoint operators>, orthonormal <singular vectors> and squared-norm <normal equation for a linear inverse problem> conditions require <Hilbert spaces>. For arbitrary <normed spaces> the <Banach-space adjoint> takes values in the dual spaces and need not produce this formula. Interpret the given <singular value system> in <Hilbert spaces>, with $u_j$ in the unknown space and $v_j$ in the data space, as in the printed identities. Positive <singular values> tending to zero provide the compact ill-posed case. A general <bounded linear operator> need not have this property; the <identity operator> is a well-conditioned counterexample. Components in the <kernel> must be fixed by a minimum-norm convention, and the unregularized inverse is defined only for data satisfying the corresponding range condition.
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