transpose operator
Let $X,Y$ be normed vector spaces^{} and ${X}^{\prime},{Y}^{\prime}$ be their continuous dual spaces.
 Let $T:X\u27f6Y$ be a bounded linear operator. The operator ${T}^{\prime}:{Y}^{\prime}\u27f6{X}^{\prime}$ given by
$${T}^{\prime}\varphi =\varphi \circ T,\varphi \in {Y}^{\prime}$$ 
is called the transpose operator of $T$ or the conjugate operator of $T$.
It is clear that ${T}^{\prime}$ is well defined, i.e. $\varphi \circ T\in {X}^{\prime}$, since the composition of two continuous linear operators is again a continuous linear operator.
Moreover, it can be easily checked that ${T}^{\prime}$ is a bounded linear operator.
Remarks 

•
When the vector spaces^{} are finite dimensional, the transpose operator corresponds to transposing (http://planetmath.org/Transpose^{}) the matrix associated to it.

•
For Hilbert spaces^{}, a somewhat similar definition is that of adjoint operator. But this two notions do not coincide: while the transpose operator corresponds to the transpose of a matrix, the adjoint operator corresponds to the conjugate transpose^{} of a matrix.
Title  transpose operator 

Canonical name  TransposeOperator 
Date of creation  20130322 17:34:19 
Last modified on  20130322 17:34:19 
Owner  asteroid (17536) 
Last modified by  asteroid (17536) 
Numerical id  5 
Author  asteroid (17536) 
Entry type  Definition 
Classification  msc 47A05 
Classification  msc 4600 
Synonym  conjugate operator 
Related topic  Transpose 
Related topic  Adjoint5 