Proof of inner product for function space

[SFB]

Hi I am kinda new to this topic two . I was wondering how can I prove that the following expressions define scalar product. All I can guess that I need to show that they follow the properties of the scalar product.

But how? If possible, help me with an example .1. (f,g)=$$\int f(x)g(x)w(x)dx$$ where w(x)>0 where x=[0,1]
2. (f,g)=$$\int f'(x)g'(x)dx$$ +f(0)g(0)

 

[Dick]

If this is homework, you should put it on the homework forums, just start a new thread. But I've already asked you, what part of the conditions for those being an inner product are you having a hard time proving?

 

[Landau]

The only non-trivial part is showing (f,f)=0 --> f=0. You will need continuity of f, so I think you forgot to give information about your function space (probably the space consisting of continuous functions f:[0,1]->R).

 

[HallsofIvy]

Show that each of those satisfies all of the conditions for an inner product. What are those conditions- that is, what is the definition of "inner product"?