Difference between revisions of "Euclidean n-space"

Latest revision as of 14:17, 12 July 2015

Definition

There are two forms of Euclidean $n$-space^[1], we have:

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

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

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

Which for real $x$ and $y$ will be recognised as the Vector dot product

Notes

Notice that the norm induced by this inner product is:

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

    If we write $x_i$ as $a_i+b_ij$ then we see that:

    • $x_i\overline{x_i}=(a_i+b_ij)(a-b_ij)=a_i^2+b_i^2$

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

Then notice that the induced metric is:

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

References

1. Jump up ↑ Functional Analysis - George Bachman and Lawrence Narici