Prove scalar product of square-integrable functions

[unscientific]

Homework Statement

Consider the vector space of continuous, complex-valued functions on the interval [−∏, ∏]. Show that

defines a scalar product on this space. Are the following functions orthogonal with respect to this scalar product?

Homework Equations

The Attempt at a Solution

Definition of scalar product: A scalar product is an operation that assigns a complex number to any given pair of vectors |a> and |b> belonging to a LVS.

Case 1: f(t) and g(t) are equal => integrand is real => scalar product is real
Case 2: f(t) and g(t) are not equal => integrand is complex => scalar product is complex

I'm not sure if this shows that the function is a scalar product. Is there a more mathematically rigorous way of showing it?

 

[micromass]

To be an inner product, you need to satisfy a few axioms. Which axioms? You need to check those.

 

[unscientific]

To be an inner product, you need to satisfy a few axioms. Which axioms? You need to check those.

Are these the axioms that are required? What's the difference between a scalar product and an inner product?

 

[micromass]

Yes, you need to check (1), (2) and (3).

A scalar product is the same as an inner product. But I prefer the latter terminology. Sorry for the confusion.

 

[unscientific]

Yes, you need to check (1), (2) and (3).

A scalar product is the same as an inner product. But I prefer the latter terminology. Sorry for the confusion.

If this operation is right, then yes it is an inner product.

How do you prove that this is right? My idea is this:

Suppose the above integrand is a complex number that contains some 'i' s, and since the process of integration is only with respect to (t), the 'i' s are unaltered. Therefore, there is no difference in replacing all the 'i' s with '-i' s before or after the process of integration. (I like to think of it as the 'i' s in the integrand retains its symmetry)

 

[micromass]

Yes, what you say is correct.