Question

Kindly answer this as soon as possible. Urgent Elaborate each step.

Show that Euclidean space and unitary space are not compact. Explain each step. 

Answer 1

Euclidian space sometimes called the cartesian space is the space of n-tuples of real numbers $(r_1, r_2,...r_n)$. It is denoted by $\reals^n$

recall that space x is said to be compact if each of its open cover has a finite subcover.

Euclidian space is not compact:

Reason: Take $0=(0,0,0,...0)\in \reals^n$

consider an open ball of radius R around o for each $k\in \N.$ denote it by $B_k(0,k).$

$\reals^n \subseteq \displaystyle\cup_{k=1}^{\infin} B_k(0,k).$ But it has no finite subcovering corresponding to the covering $\{B_k(0,k)\}K\in\N.$

Hence, it is not compact.

Recall that a unitary space is n-dimensional complex linear space on which there is an inner product.

recall that for any $r,y\in t$

$d(r,y)=\sqrt{<r-y,r-y>}$

similar to euclidian space, we may take any point $r\space t\space \in$ and construct an open cover at $E\subseteq\displaystyle\cup_{k=1}^{\infin}(n,k),$ having no finite subcover.

This gives that unitary space is also not compact.