Is <f,g> an Inner Product on C[a,b]?

[Shaunzio]

Homework Statement

show that if we define the following operation:
let f=f(x) and g=g(x) be two functions in C[a,b] and define <f,g>=int(a to b) f(x)g(x)dx
show that the conditions of therom are satisified with this operation. Use h=h(x) to help with part b
this shows that this operation is an inner product
therom

Homework Equations

a. u dot v = v dot u
b. (u+v) dot w = u dot w + v dot w
c. (cu) dot v = c(u dot v) = u dot (cv)
d. u dot u >= 0, and u dot u = 0 if and only if u = 0

The Attempt at a Solution

I really have no idea where to begin here...

 

[Sethric]

I assume you are trying to prove that your structure is an inner product. For reference, <f,g> is just a generalized way of writing a dot product (as a dot product is just one specific inner product). So what you need to prove is:

a. <f,g> = <g,f>
b. <f+g,h> = <f,h> + <g,h>
c. <cf,g> = c<f,g> = <f,cg> where c is a scalar, not a function

and so on. Plug and chug, almost.

 

[Shaunzio]

Hi sorry. I'm still pretty confused. Is there anyway you could elaborate?

 

[HallsofIvy]

Homework Statement

show that if we define the following operation:
let f=f(x) and g=g(x) be two functions in C[a,b] and define <f,g>=int(a to b) f(x)g(x)dx
show that the conditions of therom are satisified with this operation. Use h=h(x) to help with part b
this shows that this operation is an inner product
therom

It's rather confusing for us to use "<f, g>" above and "u dot v" below!

Homework Equations

a. u dot v = v dot u

So you want to prove that
$$<f, g>= \int_a^b f(x)g(x)dx= \int_a^b g(x)f(x)dx= <g, f>$$

b. (u+v) dot w = u dot w + v dot w

So you want to prove that
$$<f+ g, h>= \int_a^b (f(x)+ g(x))h(x) dx= \int_a^b f(x)h(x) dx+ \int_a^b g(x)h(x) dx= <f, h>+ <g, h>$$

c. (cu) dot v = c(u dot v) = u dot (cv)

So you want to prove that
$$<cf, g>= \int_a^b (cf(x))g(x)dx= c\int_a^b f(x)g(x)dx= \int_a^b f(x)(cg(x))dx$$

d. u dot u >= 0, and u dot u = 0 if and only if u = 0

So you want to prove that
$$<f, f>= \int_a^b f^2(x)dx\ge 0$$
and
$$<f, f>= \int_a^b f^2(x)dx= 0$$
if and only if f(x)= 0 for all x between a and b.

The Attempt at a Solution

I really have no idea where to begin here...

Did you not consider writing out what you want to prove in terms of the given definition?