Editing Dual space (section)

=== Quotient spaces and annihilators ===

Let <math>S</math> be a subset of <math>V</math>.
The '''[[annihilator (ring theory)|annihilator]]''' of <math>S</math> in <math>V^*</math>, denoted here <math>S^0</math>, is the collection of linear functionals <math>f\in V^*</math> such that <math>[f,s]=0</math> for all <math>s\in S</math>.
That is, <math>S^0</math> consists of all linear functionals <math>f:V\to F</math> such that the restriction to <math>S</math> vanishes: <math>f|_S = 0</math>.
Within finite dimensional vector spaces, the annihilator is dual to (isomorphic to) the [[orthogonal complement]].

The annihilator of a subset is itself a vector space.
The annihilator of the zero vector is the whole dual space: <math>\{ 0 \}^0 = V^*</math>, and the annihilator of the whole space is just the zero covector: <math>V^0 = \{ 0 \} \subseteq V^*</math>.
Furthermore, the assignment of an annihilator to a subset of <math>V</math> reverses inclusions, so that if <math>\{ 0 \} \subseteq S\subseteq T\subseteq V</math>, then
: <math>
    \{ 0 \} \subseteq T^0 \subseteq S^0 \subseteq V^* .
  </math>

If <math>A</math> and <math>B</math> are two subsets of <math>V</math> then
: <math>
    A^0 + B^0 \subseteq (A \cap B)^0 .
  </math>
If <math>(A_i)_{i\in I}</math> is any family of subsets of <math>V</math> indexed by <math>i</math> belonging to some index set <math>I</math>, then
: <math>
    \left( \bigcup_{i\in I} A_i \right)^0 = \bigcap_{i\in I} A_i^0 .
  </math>
In particular if <math>A</math> and <math>B</math> are subspaces of <math>V</math> then
: <math>
    (A + B)^0 = A^0 \cap B^0
  </math>
and<ref group=nb name="choice"/>
: <math>
    (A \cap B)^0 = A^0 + B^0 .
  </math>

If <math>V</math> is finite-dimensional and <math>W</math> is a [[vector subspace]], then
: <math>
    W^{00} = W
  </math>
after identifying <math>W</math> with its image in the second dual space under the double duality isomorphism <math>V\approx V^{**}</math>.  In particular, forming the annihilator is a [[Galois connection]] on the lattice of subsets of a finite-dimensional vector space.

If <math>W</math> is a subspace of <math>V</math> then the [[quotient space (linear algebra)|quotient space]] <math>V/W</math> is a vector space in its own right, and so has a dual.  By the [[first isomorphism theorem]], a functional <math>f:V\to F</math> factors through <math>V/W</math> if and only if <math>W</math> is in the [[kernel (algebra)|kernel]] of <math>f</math>.  There is thus an isomorphism
: <math> (V/W)^* \cong W^0 .</math>
As a particular consequence, if <math>V</math> is a [[direct sum of modules|direct sum]] of two subspaces <math>A</math> and <math>B</math>, then <math>V^*</math> is a direct sum of <math>A^0</math> and <math>B^0</math>.