Showing the Dual Basis is a basis

In summary, the conversation discusses the dual space, $V^*$, and its basis which is dual to the basis of $V$. It is proven that this is a basis for $V^*$, characterized by $\alpha^i(e_j)=\delta_j^i$. The Riesz representation theorem is also mentioned, which states that if $T:V \rightarrow \Bbb{R}$ is linear, then there is a vector $t$ such that $T(u) = <t,u>$, and this can be proven using the Gram-Schmidt process. The speaker is looking for an outline of this proof.
  • #1
joypav
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0
I am working through a book with my professor and we read a section on the dual space, $V^*$.
It gives the basis dual to the basis of $V$ and proves that this is in fact a basis for $V^*$.
Characterized by $\alpha^i(e_j)=\delta_j^i$

I understand the proof given. But he said a different statement...
If $T: V \rightarrow \Bbb{R}$ is linear, then there is a vector $t$ so that $T(u)=<t,u>$. ($<\cdot, \cdot>$ is the inner product)

He said this is equivalent to showing that the dual basis forms a basis, and that it can be proven using the Gram-Schmidt process. I was wondering what that proof looks like? Even just an outline... I don't need all the details.
 
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  • #2
Hi joypav,

The result your advisor mentioned is the Riesz representation theorem. There are many books, websites, and papers that you can look to for its statement and proof. Have a look at one and feel free to follow up with any questions.
 

1. What is the dual basis?

The dual basis is a set of vectors that are dual to the basis of a vector space. It consists of linear functionals that map each basis vector to its corresponding coordinate in the basis.

2. Why is it important to show that the dual basis is a basis?

Showing that the dual basis is a basis is important because it proves that the set of linear functionals is linearly independent and spans the entire vector space, making it a complete and unique basis for the dual space.

3. How do you show that the dual basis is a basis?

To show that the dual basis is a basis, you need to prove two things: linear independence and spanning. Linear independence can be shown by setting up a system of equations and solving for the coefficients of the linear functionals. Spanning can be shown by demonstrating that each vector in the original basis can be expressed as a linear combination of the dual basis vectors.

4. Can the dual basis be different from the original basis?

Yes, the dual basis can be different from the original basis. In fact, the dual basis is only defined for finite-dimensional vector spaces, and in most cases, the dual basis will be different from the original basis.

5. What is the relationship between the dual basis and the original basis?

The dual basis is a basis for the dual space, which is the set of all linear functionals on the original vector space. The dual basis is constructed using the original basis, and it provides a way to represent vectors in the original space as linear combinations of linear functionals in the dual space.

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