Proof of inner product for function space

In summary, the conversation is about proving that two expressions, (f,g)=\int f(x)g(x)w(x)dx where w(x)>0 and (f,g)=\int f'(x)g'(x)dx +f(0)g(0), define a scalar product. The person is asking for help with an example and what conditions need to be satisfied for these expressions to be considered an inner product. They mention that continuity of f is needed and that information about the function space may have been forgotten.
  • #1
SFB
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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)=[tex]\int f(x)g(x)w(x)dx[/tex] where w(x)>0 where x=[0,1]
2. (f,g)=[tex]\int f'(x)g'(x)dx[/tex] +f(0)g(0)
 
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  • #2
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?
 
  • #3
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).
 
  • #4
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"?
 

1. What is an inner product in a function space?

An inner product in a function space is a mathematical operation that takes two functions as input and produces a scalar value as output. It is a generalization of the dot product in vector spaces. The inner product can measure the similarity or orthogonality of two functions.

2. How is the inner product defined in a function space?

The inner product in a function space is defined as the integral of the product of two functions over a given domain. In other words, it is the area under the curve of the product of the two functions. This definition allows for a wide range of functions to be used in the inner product.

3. What is the significance of the inner product in function spaces?

The inner product is significant in function spaces because it allows for the definition of important concepts such as orthogonality, projection, and distance. It also provides a way to generalize geometric concepts from vector spaces to function spaces, which is useful in many applications.

4. How is the inner product used to prove properties of function spaces?

The inner product is used to prove properties of function spaces by providing a mathematical framework for analyzing and manipulating functions. By defining a set of axioms for the inner product, we can prove important properties such as the Cauchy-Schwarz inequality, the Pythagorean theorem, and the completeness of function spaces.

5. Can the inner product be extended to other types of function spaces?

Yes, the inner product can be extended to other types of function spaces such as complex-valued functions, infinite-dimensional function spaces, and general inner product spaces. The definition and properties of the inner product may vary in these spaces, but the fundamental concept remains the same.

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