if where the left hand side is also equal to The left hand side of this characterization involves only linear functionals of the form while the right hand side involves only linear functions of the form (defined as above). So in plain English, characterization () says that an operator is normal when the inner product of any two linear functions of the first form is equal to the inner product of their second form (using the same vectors for both forms). In other words, if it happens to be the case (and when is injective, it is) that the assignment of linear functionals is well-defined (or alternatively, if is well-defined) where ranges over then is a normal operator if and only if this assignment preserves the inner product on The fact that every self-adjoint bounded linear operator is normal follows readily by direct substitution of into either side of This same fact also follows immediately from the direct substitution of the equalities () into either side of (). Alternatively, for a complex Hilbert space, the continuous linear operator is a normal operator if and only if for every which happens if and only if Unitary operators An invertible bounded linear operator is said to be unitary if its inverse is its adjoint: By using (), this is seen to be equivalent to Unraveling notation and definitions, it follows that is unitary if and only if The fact that a bounded invertible linear operator is unitary if and only if (or equivalently, ) produces another (well-known) characterization: an invertible bounded linear map is unitary if and only if Because is invertible (and so in particular a bijection), this is also true of the transpose This fact also allows the vector in the above characterizations to be replaced with or thereby producing many more equalities. Similarly, can be replaced with or See also Citations Notes Proofs Bibliography P. Halmos Measure Theory, D. van Nostrand and Co., 1950. P. Halmos, A Hilbert Space Problem Book, Springer, New York 1982 (problem 3 contains version for vector spaces with coordinate systems). Walter Rudin, Real and Complex Analysis, McGraw-Hill, 1966, . Articles containing proofs Duality theories Integral representations Theorems in functional analysis