22 Norm of a vector

Let $V={𝐂}^{n\times 1}$, the column vectors. The standard inner product on $V$ is defined for $z=\text{column}{(z_j)}_{j=1}^n$ $w=\text{column}{(w_j)}_{j=1}^n$ by

$$⟨z,w⟩={\overline{w}}^{T}z=\sum _{j=1}^n{z}_j{\overline{w}}_j.$$

Then

$$⟨z+u,w⟩=⟨z,w⟩+⟨u,w⟩$$

$$⟨\lambda z,w⟩=\lambda ⟨z,w⟩,\overline{⟨z,w⟩}=⟨w,z⟩$$

On $V$ the standard norm is the Euclidean norm for $z=\text{column}{(z_j)}_{j=1}^n$

$$\parallel {(z_j)}_{j=1}^n\parallel ={\left(\sum _{j=1}^n{|z_j|}^2\right)}^{1/2}={({\overline{z}}^{T}z)}^{1/2}={⟨z,z⟩}^{1/2}.$$