Editing Dual space (section)

=== Transpose of a continuous linear map ===
{{See also|Transpose of a linear map|Dual system#Transposes}}

If {{math|''T'' : ''V → W''}} is a continuous linear map between two topological vector spaces, then the (continuous) transpose  {{math|''T′'' : ''W′ → V′''}} is defined by the same formula as before:

:<math>T'(\varphi) = \varphi \circ T, \quad \varphi \in W'.</math>

The resulting functional {{math|''T′''(''φ'')}} is in {{math|''V′''}}. The assignment {{math|''T → T′''}} produces a linear map between the space of continuous linear maps from ''V'' to ''W'' and the space of linear maps from {{math|''W′''}} to {{math|''V′''}}.
When ''T'' and ''U'' are composable continuous linear maps, then

:<math>(U \circ T)' = T' \circ U'.</math>

When ''V'' and ''W'' are normed spaces, the norm of the transpose in{{math| ''L''(''W′'', ''V′'')}} is equal to that of ''T'' in {{math|''L''(''V'', ''W'')}}.
Several properties of transposition depend upon the [[Hahn–Banach theorem]].
For example, the bounded linear map ''T'' has dense range if and only if the transpose {{math|''T′''}} is injective.

When ''T'' is a [[Compact operator|compact]] linear map between two Banach spaces ''V'' and ''W'', then the transpose {{math|''T′''}} is compact.
This can be proved using the [[Arzelà–Ascoli theorem]].

When ''V'' is a Hilbert space, there is an antilinear isomorphism ''i<sub>V</sub>'' from ''V'' onto its continuous dual {{math|''V′''}}.
For every bounded linear map ''T'' on ''V'', the transpose and the [[Hermitian adjoint|adjoint]] operators are linked by

:<math>i_V \circ T^* = T' \circ i_V.</math>

When ''T'' is a continuous linear map between two topological vector spaces ''V'' and ''W'', then the transpose {{math|''T′''}} is continuous when {{math|''W′''}} and {{math|''V′''}} are equipped with "compatible" topologies: for example, when for {{math|1=''X'' = ''V''}} and {{math|1=''X'' = ''W''}}, both duals {{math|''X′''}} have the [[Strong topology (polar topology)|strong topology]] {{math|''β''(''X′'', ''X'')}} of uniform convergence on bounded sets of ''X'', or both have the weak-∗ topology {{math|''σ''(''X′'', ''X'')}} of pointwise convergence on&nbsp;''X''.
The transpose {{math|''T′''}} is continuous from {{math|''β''(''W′'', ''W'')}} to {{math|''β''(''V′'', ''V'')}}, or from {{math|''σ''(''W′'', ''W'')}} to {{math|''σ''(''V′'', ''V'')}}.