The hermiticity conditions are not invariant under the action \ gamma ^ \ mu \ to S ( \ Lambda ) \ gamma ^ \ mu { S ( \ Lambda ) } ^ {-1 } of a Lorentz transformation \ Lambda because S ( \ Lambda ) is not necessarily a unitary transformation due to the noncompactness of the Lorentz group.
42.
By spectral theory, there is an essentially unique measure ? on the locally compact space " X " and a unitary transformation " U " between " H " 0 and " L " 2 ( " X ", ? ) which carries the operators in \ mathfrak { A } onto the corresponding multiplication operators.
43.
With the above ordering of the basic vectors, the unitary transformations from the basis ( a'_ i ) to the basis ( a _ i ) and from the basis ( b'_ i ) to the basis ( b _ i ) realise a singular value decomposition of the matrix of inner products \ langle a'_ i, b'_ j \ rangle.
44.
Pre-composing " g " with a scaling transformation and post-composing " g " with a unitary transformation, it can be assumed that " g " ( 0 ) is a diagonal matrix with entries ? " i " e " 0 with " r " = ? 1 > 0 and that the image of the unit ball is contained in a small ball around this point.
45.
By the uniqueness of the singular value decomposition, the vectors \ hat y _ i are then unique up to a real, orthogonal or unitary transformation among them, and the vectors \ hat w _ i and \ hat w'_ i ( and hence \ hat v _ i ) are unique up to equal real, orthogonal or unitary transformations applied simultaneously to the sets of the vectors \ hat w _ i associated with a common value of \ theta _ i and to the corresponding sets of vectors \ hat w'_ i ( and hence to the corresponding sets of \ hat v _ i ).
46.
By the uniqueness of the singular value decomposition, the vectors \ hat y _ i are then unique up to a real, orthogonal or unitary transformation among them, and the vectors \ hat w _ i and \ hat w'_ i ( and hence \ hat v _ i ) are unique up to equal real, orthogonal or unitary transformations applied simultaneously to the sets of the vectors \ hat w _ i associated with a common value of \ theta _ i and to the corresponding sets of vectors \ hat w'_ i ( and hence to the corresponding sets of \ hat v _ i ).
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