Verifying Inner Product Space: q(x)e^-(x^2/2)

In summary, if one has functions of the form q(x)e^-(x^2/2) where q(x) is a polynomial of degree < N in x, on the interval -∞ < x < ∞, then they can verify if they form an inner product space by checking if the integral of the product of any two such functions is finite. Additionally, they can specify the dimension of the space by checking if the set of all equivalence classes satisfies the condition that the elements of the set are functions almost everywhere.
  • #1
Thunder_Jet
18
0
Hi everyone!

I would like to ask how would you verify if functions form an inner product space? For example, if one has functions of the form q(x)e^-(x^2/2) where q(x) is a polynomial of degree < N in x, on the interval -∞ < x < ∞. Also, how would you specify the dimension of the space, if it exists?

Thank you!
 
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  • #2
The dimension is N. It forms an inner product space, since the integral of the product of any two such functions is finite (the exponential term insures this).
 
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  • #3
Check using the definition of an inner product would be my initial suggestion.

1) [itex](u|u)\geq 0[/itex] and [itex]0[/itex] iff [itex]u = 0[/itex]
2) [itex](\alpha u+ \beta v|w) = \alpha (u|w) + \beta (v|w) [/itex] Aka that it is linear

If this holds true [itex]\forall u,v,w[/itex] then the inner product is defined for the said space and is thus a inner product space (given of course that it is in a vector space to begin with)
 
  • #4
Note that vector spaces whose elements are functions usually fail the condition [itex]\langle f,f\rangle=0\Rightarrow 0[/itex]. (Define f by f(x)=1 when x=0 and f(x)=0 otherwise; then [itex]\langle f,f\rangle=0[/itex] but f≠0). Such a space is sometimes called a semi-inner product space.
 
  • #5
Fredrik said:
Note that vector spaces whose elements are functions usually fail the condition [itex]\langle f,f\rangle=0\Rightarrow 0[/itex]. (Define f by f(x)=1 when x=0 and f(x)=0 otherwise; then [itex]\langle f,f\rangle=0[/itex] but f≠0). Such a space is sometimes called a semi-inner product space.

Since f = 0 almost everywhere, this will be an inner product space. Moreover it is complete, so it is L2.
 
  • #6
mathman said:
Since f = 0 almost everywhere, this will be an inner product space.
By definition of inner product, it's not. But you can define an equivalence relation by saying that f~g if f=g almost everywhere, and then define an inner product on the set of equivalence classes by [itex]\langle [f],[g]\rangle=\langle f,g\rangle[/itex]. That's an inner product on the left and a semi-inner product on the right.
 
  • #7
Thank you so much for the insights! All of your comments gave me an idea on how to attack the problem! Thanks once again!
 

1. What is an inner product space?

An inner product space is a mathematical concept used in linear algebra to define the notion of "length" or "magnitude" of a vector, as well as the angle between two vectors. It is a vector space equipped with an inner product, which is a function that takes in two vectors and returns a scalar value.

2. How do you verify an inner product space?

In order to verify that a vector space is an inner product space, we need to check that it satisfies the three axioms of an inner product: linearity in the first argument, conjugate symmetry, and positive-definiteness. This means that the inner product must be linear in the first argument, satisfy the property of conjugate symmetry (where the inner product of two vectors is equal to the complex conjugate of the inner product of the same two vectors in reverse order), and have a positive-definite inner product matrix.

3. What is the purpose of verifying an inner product space?

The purpose of verifying an inner product space is to ensure that the vector space has the necessary properties to define a notion of "length" and "angle" between vectors. This is important in many areas of mathematics, as well as in physics and engineering, where vector spaces are used to model physical quantities.

4. How does q(x)e^-(x^2/2) relate to verifying an inner product space?

q(x)e^-(x^2/2) is a specific inner product function that is commonly used in the verification of inner product spaces. This function is known as the Gaussian inner product and is often used in probability and statistics. By verifying that this function satisfies the three axioms of an inner product, we can determine if a vector space is an inner product space.

5. What are some applications of inner product spaces?

Inner product spaces have many applications in mathematics, physics, and engineering. They are used to define norms and distances, which are important in optimization problems and numerical analysis. They are also used in Fourier analysis, where they help in the decomposition of functions into their frequency components. In addition, inner product spaces play a crucial role in quantum mechanics, where they are used to define the concept of "orthogonality" between quantum states.

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