Simple proof of Complex Inner Product Space

In summary, the conversation discusses proving that the inner product of any vector |v> with the zero vector equals 0 in a general inner product space. The proof is divided into two cases, one where |v> is equal to 0 and one where it is not. The proof for the first case is straightforward due to an axiom of inner spaces. For the second case, the summation definition of the inner product is used to show that the sum of the components of <v|0> equals 0. The questions asked pertain to the validity and correctness of the proof. Ultimately, it is determined that the proof is correct and the scalar 0 is proven to be the result, rather than the vector 0.
  • #1
RJLiberator
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Homework Statement


Prove that <v|0>=0 for all |v> ∈ V.

Homework Equations

The Attempt at a Solution



This is a general inner product space.

I break it up into 2 cases.
Case 1: If |v> = 0, the proof is trivial due to inner space axiom stating <0|0> = 0.

Case 2: If |v> =/= 0 then:
I use <v|0> = Σv_i * 0_i
and from here it is clear to see that the sum adds up to 0 as every component is multiplied by the 0 vector.

My question: Is this a safe definition of the complex inner product? Am I OK to use the summation definition in this general proof?
Second Question: Is the proof correct? Any reason why the 0 vector would need to be proven further to sum the components to 0?

Thanks
 
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  • #2
I think I have a better idea.

<v|0> = <v|0v> (by multiplication by 0.)
= 0<v|v> by inner product axiom 2
=0
and done!

This is the scalar 0, and not the vector 0 as stated in the question.
 

1. What is a complex inner product space?

A complex inner product space is a vector space over the field of complex numbers that also has an inner product defined. This means that for any two vectors in the space, there is a unique complex number that represents the magnitude and direction of their inner product.

2. How is a complex inner product space different from a real inner product space?

A complex inner product space differs from a real inner product space in that the inner product in a complex space can take on complex values, while in a real space it is restricted to real values. This allows for a more flexible and powerful representation of vectors in a complex space.

3. What is the significance of a simple proof of complex inner product space?

A simple proof of complex inner product space is significant because it provides a clear and concise way of understanding and verifying the properties and characteristics of this type of vector space. It also allows for easier application and development of complex inner product spaces in various fields of study.

4. Can you provide an example of a complex inner product space?

One example of a complex inner product space is the space of polynomials with complex coefficients. The inner product in this space is defined as the integral of the product of two polynomials over a given interval. This space satisfies all the properties of a complex inner product space.

5. What are some applications of complex inner product space?

Complex inner product spaces have many applications in mathematics, physics, and engineering. They are used in quantum mechanics to represent states of particles, in signal processing to analyze and manipulate complex signals, and in computer graphics to create and manipulate complex images. They are also used in various branches of mathematics, such as functional analysis and differential geometry.

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