structural:expansion, structural:scholar_edition, stylistic:tone, information:filtering
Finite-dimensional complex normed spaces

On an n {\displaystyle n} -dimensional complex space C n, {\displaystyle \mathbb {C} ^{n},} the most common norm is ‖ z ‖:= | z 1 | 2 + ⋯ + | z n | 2 = z 1 z ¯ 1 + ⋯ + z n z ¯ n. {\displaystyle \|{\boldsymbol {z}}\|:={\sqrt {\left|z_{1}\right|^{2}+\cdots +\left|z_{n}\right|^{2}}}={\sqrt {z_{1}{\bar {z}}_{1}+\cdots +z_{n}{\bar {z}}_{n}}}.}

In this case, the norm can be expressed as the square root of the inner product of the vector and itself: ‖ x ‖:= x H x, {\displaystyle \|{\boldsymbol {x}}\|:={\sqrt {{\boldsymbol {x}}^{H}~{\boldsymbol {x}}}},} where x {\displaystyle {\boldsymbol {x}}} is represented as a column vector T {\displaystyle {\begin{bmatrix}x_{1}\;x_{2}\;\dots \;x_{n}\end{bmatrix}}^{\rm {T}}} and x H {\displaystyle {\boldsymbol {x}}^{H}} denotes its conjugate transpose.

This formula is valid for any inner product space, including Euclidean and complex spaces. For complex spaces, the inner product is equivalent to the complex dot product. Hence the formula in this case can also be written using the following notation: ‖ x ‖:= x ⋅ x. {\displaystyle \|{\boldsymbol {x}}\|:={\sqrt {{\boldsymbol {x}}\cdot {\boldsymbol {x}}}}.}