ORTHOGONAL COMPLEMENT

In the mathematical fields of linear algebra and functional analysis, the 'orthogonal complement' of a subspace ''W'' of an inner product space ''V'' is the set of all vectors in ''V'' that are orthogonal to every vector in ''W'', i.e., it is
:
angle = 0 ,
ight}.,
The orthogonal complement is always closed in the metric topology. In Hilbert spaces, the orthogonal complement of the orthogonal complement of ''W'' is the closure of ''W'', i.e.,
:
If A is an $m\; imes\; n$ matrix, $mbox\{Row\; \}\; A$, $\{Col\; \}\; A$, and $mbox\{Nul\; \}\; A$ refer to the row space, column space, and null space of A (respectively), we have
:
and
:

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Banach spaces

Banach spaces

There is a natural analog of this notion in general Banach spaces. In this case one defines the orthogonal complement of ''W'' to be a subspace of the dual of ''V'' defined similarly by
:
★ : orall yin W x(y) = 0 ,
ight}.,
It is always a closed subspace of $V^$
★ . There is also an analog of the double complement property. is now a subspace of $\{V^$
★ }^
★ (which is not identical to $V$). However, the reflexive spaces have a natural isomorphism $i$ between $V$ and $\{\{V^$
★ }^
★ }. In this case we have
:
This is a rather straightforward consequence of the Hahn-Banach theorem.