Let *T* : *B*_{1} → *B*_{2} be an operator between Banach spaces. Then the *transpose* (or *dual*) of *T* is an operator satisfying:

for all *x* in B_{1} and *y* in B_{2}*. Here, we used the notation: .

The necessary and sufficient condition for the transpose of *T* to exist is that *T* is densely defined (for essentially the same reason as to adjoints, as discussed above.)

For any Hilbert space *H*, there is the anti-linear isomorphism:

given by where . Through this isomorphism, the transpose *T*' relates to the adjoint *T*∗ in the following way:

- ,

where . (For the finite-dimensional case, this corresponds to the fact that the adjoint of a matrix is its conjugate transpose.) Note that this gives the definition of adjoint in terms of a transpose.

Read more about this topic: Closed Operator

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