Becall AccC X,ch) AB :bounded ifJACX,r yo Sucb thaAc BCXX)property (Xdl) metric epaceAc X, A Compact in CX, Ja J A isboundedproof.in (X,d)Hint tetie x carbitraryIdnd BX,n).nEl, 2,3.Show opan Cover of XA CompuctinuSGcondlude Definition: A space X is compact provided that every open cover of X has a finitesubcover. Equivalently, X is compact provided that for every collection O of opensets whose union equals X, there is a finite subcollection {O;}\1 of O whose unionequals X. A subspace A of a space X is compact provided that A is a compacttopological space in its subspace topology.

We will use the definition of compact set and bounded set to prove this property.

-Re call, Ac (Xicd) AB .bounded if JxcX,ryo Sucb that Ac BCX) A B(メイ) property (Xdl) methic space AcX, A Compact insX, JaJ → bounded A is prof. in (X,d) Hint: tette x crbitrary in X Idnd BX,n).n=1, 2,3. open Cover of X A Compuct Conclude # Show

-Re call, Ac (Xicd) AB .bounded if JxcX,ryo Sucb that Ac BCX) A B(メイ) property (Xdl) methic space AcX, A Compact insX, JaJ → bounded A is prof. in (X,d) Hint: tette x crbitrary in X Idnd BX,n).n=1, 2,3. open Cover of X A Compuct Conclude # Show

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter4: Vector Spaces

Section4.2: Vector Spaces

Problem 37E: Let V be the set of all positive real numbers. Determine whether V is a vector space with the...

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Concept explainers

Linear Algebra

In mathematics, problems can be solved by the arithmetic operators like addition, subtraction, multiplication, and division. In algebra, we represent the numbers by any letter called a variable. If these variables are of degree 1, then it is studied under linear algebra. The first-degree equations involving algebraic expressions are called linear equations. That is, if you represent it by drawing a graph you will get a straight line.

Question

Becall AccC X,ch) AB :bounded if
JACX,r yo Sucb tha
Ac BCXX)
property (Xdl) metric epace
Ac X, A Compact in CX, Ja J A is
bounded
proof.
in (X,d)
Hint tetie x carbitrary
Idnd BX,n).nEl, 2,3.
Show opan Cover of X
A Compuct
in
uSG
condlude

Definition: A space X is compact provided that every open cover of X has a finite
subcover. Equivalently, X is compact provided that for every collection O of open
sets whose union equals X, there is a finite subcollection {O;}\1 of O whose union
equals X. A subspace A of a space X is compact provided that A is a compact
topological space in its subspace topology.

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