Posted by **Tri N** on Sunday, March 6, 2011 at 2:14pm.

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)

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