Euclidean n-space

Definition

There are two forms of Euclidean [ilmath]n[/ilmath]-space, we have:

1. Complex Euclidean [ilmath]n[/ilmath]-space and
   • Defined by [ilmath]X=\mathbb{C}^n[/ilmath] (so [ilmath]X[/ilmath] consists of all [ilmath]n[/ilmath]-tuples of the form [ilmath](x_1,\cdots,x_n)[/ilmath] where [ilmath]x_i\in\mathbb{C} [/ilmath])
2. Real Euclidean [ilmath]n[/ilmath]-space
   • Defined by [ilmath]X=\mathbb{R}^n[/ilmath] (so [ilmath]X[/ilmath] consists of all [ilmath]n[/ilmath]-tuples of the form [ilmath](x_1,\cdots,x_n)[/ilmath] where [ilmath]x_i\in\mathbb{R} [/ilmath])

Both of these are inner-product spaces, equipped with the same inner product, being:

• [ilmath]\forall x,y\in X[/ilmath] we define the inner product as:
  • [ilmath]\langle x,y\rangle := \sum^n_{i=1}x_i\overline{y_i}[/ilmath]

Which for real [ilmath]x[/ilmath] and [ilmath]y[/ilmath] will be recognised as the Vector dot product

Notes

Notice that the norm induced by this inner product is:

• [ilmath]\forall x\in X[/ilmath] we define the norm as: [ilmath]\Vert x\Vert:=\sqrt{\langle x,x\rangle}[/ilmath]
  • Note: that means [ilmath]\Vert x\Vert=\sqrt{\sum^n_{i=1}x_i\overline{x_i} }[/ilmath]

    If we write [ilmath]x_i[/ilmath] as [ilmath]a_i+b_ij[/ilmath] then we see that:

    • [ilmath]x_i\overline{x_i}=(a_i+b_ij)(a-b_ij)=a_i^2+b_i^2[/ilmath]

  • Thus [ilmath]\Vert x\Vert =\sqrt{\sum^n_{i=1}(a_i^2+b_i^2)}[/ilmath]
    • If the [ilmath]x_i\in\mathbb{R} [/ilmath] then this is the usual length of a vector in [ilmath]\mathbb{R}^n[/ilmath]
    • If the [ilmath]x_i\in\mathbb{C} [/ilmath] then this is still the usual length of a (complex) vector (to see this use the case [ilmath]n=1[/ilmath] and see a complex vector as a vector in [ilmath]\mathbb{R}^2[/ilmath])

Then notice that the induced metric is:

• [ilmath]\forall x,y\in X[/ilmath] defined by [ilmath]d(x,y)=\Vert x-y\Vert[/ilmath]
  • Thus [math]d(x,y)=\sqrt{\sum^n_{i=1}(x-y)\overline{(x-y)} }=\sqrt{\sum_{i=1}^n\left((x_r-y_r)^2+(x_i-y_i)^2\right) }[/math]
    • And again, if [ilmath]x,y\in\mathbb{R} [/ilmath] then the imaginary component is zero and this is the Euclidean distance the reader ought to be familiar with