Proof of inner product for function space

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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)
 
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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?
 
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).
 
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"?
 

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