# A Couple of Vector Space Problems

• Mar 5th 2011, 02:48 PM
huutri030489
A Couple of Vector Space Problems
1/ Prove that the set V=R+ ( the set of all positive real numbers) is a vector space with the following nonstandard operations: for any x,y belong to R+ & for any scalar c belong to R:
x O+ ( +signal into circle) y=x.y (definition of vector addition) & c O ( dot signal into circle) x = x^c (definition of scalar multiplcation) ( must verify that all 10 axioms defining a vectorspace are satisfied).
2/ Consider the vector space V = C (-infinite,infinite)= all fumctions f(x) which are continuous everywhere. Show that the following subset H of C (-infinite, infinite) is in fact a subspace of C (-infinite,infinite):
H= {all functions f(x) satisfying the differential equation f " (x) +25 f(x)=0}
(need to verify all 3 subspace requirements)
• Mar 5th 2011, 06:14 PM
tonio
Quote:

Originally Posted by huutri030489
1/ Prove that the set V=R+ ( the set of all positive real numbers) is a vector space with the following nonstandard operations: for any x,y belong to R+ & for any scalar c belong to R:
x O+ ( +signal into circle) y=x.y (definition of vector addition) & c O ( dot signal into circle) x = x^c (definition of scalar multiplcation) ( must verify that all 10 axioms defining a vectorspace are satisfied).

A v.s. must fulfil distributivity: \$\displaystyle c\odot (x+y)=c\odot x+c\odot y\$ ...is this true in this case?

Tonio

2/ Consider the vector space V = C (-infinite,infinite)= all fumctions f(x) which are continuous everywhere. Show that the following subset H of C (-infinite, infinite) is in fact a subspace of C (-infinite,infinite):
H= {all functions f(x) satisfying the differential equation f " (x) +25 f(x)=0}
(need to verify all 3 subspace requirements)

.