Answer to Question #250539 in Functional Analysis for Zia

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.