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)

1/ To prove that the set V = R+ (the set of all positive real numbers) with the defined operations is a vector space, we need to verify that all 10 axioms defining a vector space are satisfied.

1. Closure under addition: For any x, y in R+, we defined vector addition as x O+ y = x * y. Since the product of two positive real numbers is also a positive real number, the operation x O+ y = x * y is closed under the set R+.

2. Commutativity of addition: For any x, y in R+, we have x O+ y = x * y = y * x = y O+ x.

3. Associativity of addition: For any x, y, z in R+, we have (x O+ y) O+ z = (x * y) * z = x * (y * z) = x O+ (y O+ z).

4. Identity element of addition: The identity element is 1, since for any x in R+, we have x O+ 1 = x * 1 = x.

5. Inverse elements of addition: There are no inverse elements for addition in this set, since transforming a positive number into another positive number using multiplication is not possible.

6. Closure under scalar multiplication: For any x in R+ and any scalar c in R, we defined scalar multiplication as c O (dot signal into circle) x = x^c. Since raising a positive real number to any real power still results in a positive real number, the operation c O (dot signal into circle) x = x^c is closed under the set R+.

7. Associativity of scalar multiplication: For any c1, c2 in R and any x in R+, we have (c1 * c2) O (dot signal into circle) x = x^(c1 * c2) = (x^c1)^c2 = c1 O (dot signal into circle) (c2 O (dot signal into circle) x).

8. Identity element of scalar multiplication: The identity element is 1, since for any x in R+, we have 1 O (dot signal into circle) x = x^1 = x.

9. Distributivity of scalar multiplication with respect to vector addition: For any c in R and any x, y in R+, we have c O (dot signal into circle) (x O+ y) = x^c * y^c = (x^c) O+ (y^c).

10. Distributivity of scalar multiplication with respect to scalar addition: For any c1, c2 in R and any x in R+, we have (c1 + c2) O (dot signal into circle) x = x^(c1 + c2) = x^c1 * x^c2 = (c1 O (dot signal into circle) x) O+ (c2 O (dot signal into circle) x).

Since all 10 axioms are satisfied, we can conclude that the set V = R+ with the defined operations is a vector space.

2/ To show that the subset H = {all functions f(x) satisfying the differential equation f"(x) + 25f(x) = 0} is a subspace of the vector space V = C (-∞, ∞) (all functions f(x) which are continuous everywhere), we need to verify the three subspace requirements.

1. Closure under addition: Suppose f1(x) and f2(x) are functions in H. We need to show that their sum f(x) = f1(x) + f2(x) also satisfies the given differential equation f"(x) + 25f(x) = 0. Taking the second derivative of f(x), we have f''(x) = f1''(x) + f2''(x). Substituting f(x) and its derivatives into the differential equation, we get (f1''(x) + f2''(x)) + 25(f1(x) + f2(x)) = 0. Since both f1(x) and f2(x) satisfy the differential equation, we have f1''(x) + 25f1(x) = 0 and f2''(x) + 25f2(x) = 0. Therefore, their sum f(x) = f1(x) + f2(x) also satisfies the differential equation f''(x) + 25f(x) = 0.

2. Closure under scalar multiplication: Suppose f(x) is a function in H and c is a scalar. We need to show that the scalar multiple cf(x) still satisfies the given differential equation f"(x) + 25f(x) = 0. Taking the second derivative of cf(x), we have (cf(x))'' = c(f''(x)). Substituting cf(x) and its derivatives into the differential equation, we get (c(f''(x))) + 25(cf(x)) = 0. Since f(x) satisfies the differential equation f''(x) + 25f(x) = 0, we have f''(x) + 25f(x) = 0. Therefore, the scalar multiple cf(x) also satisfies the differential equation f''(x) + 25f(x) = 0.

3. Contains the zero vector: The zero vector in this vector space is the function f(x) = 0, which satisfies f''(x) + 25f(x) = 0. Hence, the zero vector is in the subset H.

Since all three requirements are satisfied, we can conclude that the subset H = {all functions f(x) satisfying the differential equation f"(x) + 25f(x) = 0} is a subspace of the vector space V = C (-∞, ∞).