The Banach adjoint 
of a bounded operator 
between Banach spaces
is defined by
 |
for every continuous linear functional 
and every 
. Thus the Banach adjoint reverses the direction
of the original operator and acts between the continuous
dual spaces. It is a bounded operator with
the same operator norm,
 |
This equality follows from the Hahn-Banach theorem (Schenker 2008, p. 26-2).
For finite-dimensional 
and 
, the matrix of the Banach
adjoint in the corresponding dual bases is the transpose
of the original matrix, including over 
. This is distinct from the matrix
of the adjoint operator between Hilbert
spaces, which is the conjugate transpose
in orthonormal bases (Schenker 2008, p. 33-1).
For a densely defined linear operator 
, the domain
of 
consists of the 
for which 
extends to a continuous linear functional on 
. The density of 
makes this extension unique, and 
is this extension.
