A _Complex [vector spaces](Vector%20spaces.md)_ is a vector space whose [dual space](Complex%20vector%20spaces.md#Complex%20dual%20spaces) is a [linear map](Linear%20map.md) onto the field, $\mathbb{C}.$ [finite dimensional](Finite%20dimensional%20vector%20spaces.md) [complex vector spaces](Complex%20vector%20spaces.md) contain vectors with complex elements and are acted upon by matrices containing complex elements. %%matrix operators on complex vector spaces: $\mbox{M}_n(\mathbb{C})$, of $n \times n$ matrices with elements on $\mathbb{C}$.%% # Complex dual spaces  For a [complex vector space,](Complex%20vector%20spaces.md) $\mathcal{V},$ this means that for every element $|u\rangle \in \mathcal{V}$ there exists an element $\langle u|\in\mathcal{V}^*$ where $\langle u|=|u\rangle^\dagger.$ The $^\dagger$ is referred to as the _[adjoint](Adjoint.md) operator_. For all $|v\rangle \in \mathcal{V}$, $\langle u |$ is a [linear map](Linear%20map.md) that maps $\mathcal{V}$ onto $\mathbb{C}$. # Complex inner product spaces The [complex vector space](Complex%20vector%20spaces.md), $\mathcal{V},$ is said to be an _[complex inner product space](Complex%20vector%20spaces.md#Complex%20inner%20product%20spaces)_ if that vector space is also equipped with an [complex inner product](Inner%20products.md#Complex%20inner%20products). For vectors $|x\rangle,\,|y\rangle \in \mathcal{V},$ we define the inner product as   ^363959  ## Examples of complex inner product spaces * [Hausdorff pre-Hilbert Space](Hausdorff%20pre-Hilbert%20Space.md)s * [Hilbert Spaces](Hilbert%20Space.md) # Vector notation in complex vector spaces Note that we use bra-ket notation for all vectors in complex vector spaces. (see [vector notation](Linear%20Algebra%20and%20Matrix%20Theory%20(index).md#Vector%20notation)). #MathematicalFoundations/Algebra/AbstractAlgebra/LinearAlgebra/VectorSpaces