Editing Functional (mathematics) (section)

===Inner product spaces===
Given an [[inner product space]] <math>X,</math> and a fixed vector <math>\vec{x} \in X,</math> the map defined by <math>\vec{y} \mapsto \vec{x} \cdot \vec{y}</math> is a linear functional on <math>X.</math> The set of vectors <math>\vec{y}</math> such that <math>\vec{x}\cdot \vec{y}</math> is zero is a vector subspace of <math>X,</math> called the ''null space'' or ''[[Kernel (linear algebra)|kernel]]'' of the functional, or the [[orthogonal complement]] of <math>\vec{x},</math> denoted <math>\{\vec{x}\}^\perp.</math>

For example, taking the inner product with a fixed function <math>g \in L^2([-\pi,\pi])</math> defines a (linear) functional on the [[Hilbert space]] <math>L^2([-\pi,\pi])</math> of square integrable functions on <math>[-\pi,\pi]:</math>
<math display=block>f \mapsto \langle f,g \rangle = \int_{[-\pi,\pi]} \bar{f} g</math>