Is Inner Product of C[-1,2] & V Defined?

In summary, the conversation discusses the definitions and properties of an inner product in two different spaces: C[-1,2] and V. In the first space, the inner product <f,g> is defined as the integral of the absolute value of the sum of two functions over the interval [-1,2]. The discussion concludes that this is not an inner product because it does not satisfy the property <f+g, h> = <f,h> + <g,h> for some f and g. In the second space, the inner product <f,g> is defined as the product of the evaluations of two functions at -π plus the integral of the second derivative of the functions over the interval [-π,π]. The conversation
  • #1
daniel_i_l
Gold Member
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1) C[-1,2] is a space of all continues functions f: [-1,2] -> C (complex)
Is:
[tex] <f,g> = \int_{-1}^{2}|f(t) + g(t)|dt [/tex]
an inner product of C[-1,2]?
I think that the answer is no because:
[tex] <f+g, h> \neq <f,h> + <g,h>[/tex]
for some f and g. this can happen when all the functions are positive and so:
|f(t) + h(t) + g(t)| doesn't equal |f(t) + h(t)| + |g(t) + h(t)|

2)
V is a space of all real functions with defined double derivatives in the interval
[tex] [-\pi, \pi] [/tex]
we define:
[tex] <f,g> = f(-\pi)g(-\pi) + \int_{-\pi}^{\pi}f''(x)g''(x)dx [/tex]
is <f,g> an inner product of V?
also here i think that the answer is no because <f,f> can equal zero even if all of f isn't 0, this can happen if f(-pi) is zero and the double derivative is 0 everywere (contiues slope).
am i correct?
Thanks.
 
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  • #2
2) there is nothing in the definition of inner product that means <f,f>=0 if and only if f=0.

Wht is the definition of an inner product? do these satisfy, or not satisfy the definitions?
 
  • #3
matt grime said:
2) there is nothing in the definition of inner product that means <f,f>=0 if and only if f=0.

Isn't that the 3rd property of the inner product? look here for example:
http://planetmath.org/encyclopedia/InnerProduct.html

did i misunderstand something?

and is (1) correct?
Thanks.
 
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  • #4
Sorry, my mistake - I was thinking of a bilinear pairing, not an inner product.

So, you have the definitions, and the counter examples: what was the question?
 
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  • #5
I just wanted to make sure that my counter examples were correct.
Thanks
 

1. What is an inner product?

An inner product is a mathematical operation that takes two vectors and produces a scalar quantity. It is a generalization of the dot product in Euclidean space.

2. What is the difference between C[-1,2] and V?

C[-1,2] is a set of continuous functions defined on the interval [-1,2], while V is a vector space. The inner product between these two sets is defined as the integral of the product of the functions over the interval.

3. How is the inner product of C[-1,2] and V calculated?

The inner product of C[-1,2] and V is calculated by integrating the product of two continuous functions over the interval [-1,2]. This can be written as <f,g> = ∫-12 f(x)g(x) dx.

4. Can the inner product of C[-1,2] and V be negative?

Yes, the inner product of C[-1,2] and V can be negative. This can happen when the two functions being multiplied have opposite signs over the interval [-1,2]. In this case, the integral will result in a negative value.

5. What is the significance of the inner product in mathematics?

The inner product has many applications in mathematics, such as in vector calculus, functional analysis, and quantum mechanics. It allows us to define the concept of orthogonality, which is important in many areas of mathematics and physics.

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