418 CHAPTER 17. INNER PRODUCT SPACES

6. If you only know that ⟨u,u⟩ ≥ 0 along with the other axioms of the inner product andif you define |z| the same way, how do the conclusions of Theorem 17.1.7 change?

7. In an inner product space, an open ball is the set

B(x,r)≡ {y : |y−x|< r} .

If z ∈ B(x,r) , show there exists δ > 0 such that B(z,δ ) ⊆ B(x,r). In words, thissays that an open ball is open. Hint: This depends on the triangle inequality.

8. Let V be the real inner product space consisting of continuous functions defined on[−1,1] with the inner product given by∫ 1

−1f (x)g(x)dx

Show that{

1,x,x2}

are linearly independent and find an orthonormal basis for thespan of these vectors.

9. A regular Sturm Liouville problem involves the differential equation for an un-known function of x which is denoted here by y,(

p(x)y′)′+(λq(x)+ r (x))y = 0, x ∈ [a,b]

and it is assumed that p(t) ,q(t)> 0 for any t along with boundary conditions,

C1y(a)+C2y′ (a) = 0C3y(b)+C4y′ (b) = 0

whereC2

1 +C22 > 0, and C2

3 +C24 > 0.

There is an immense theory connected to these important problems. The constant λ

is called an eigenvalue. Show that if y is a solution to the above problem correspond-ing to λ = λ 1 and if z is a solution corresponding to λ = λ 2 ̸= λ 1, then∫ b

aq(x)y(x)z(x)dx = 0. (17.9)

Hint: Do something like this:(p(x)y′

)′ z+(λ 1q(x)+ r (x))yz = 0,(p(x)z′

)′ y+(λ 2q(x)+ r (x))zy = 0.

Now subtract and either use integration by parts or show(p(x)y′

)′ z− (p(x)z′)′ y = ((p(x)y′

)z−(

p(x)z′)

y)′

and then integrate. Boundary conditions to show that y′ (a)z(a)− z′ (a)y(a) = 0 andy′ (b)z(b)− z′ (b)y(b) = 0.