The Dual Space and Covectors .... Browdwer, Theorem 12.2 and Corollaries ....

In summary, Corollary 12.4 states that for any nonzero element in a vector space $V$, there exists an element in its dual space $V^*$ such that its evaluation on the nonzero element is not equal to zero. This can be proven by showing that the basis of $V$ has a corresponding dual basis in $V^*$, and that one of the elements in this dual basis will have a nonzero evaluation on the given nonzero element in $V$.
  • #1
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I am reading Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ...

I am currently reading Chapter 12: Multilinear Algebra and am specifically focused on Section 12.1: Vectors and Tensors ...

I need help in fully understanding Corollary 12.4 to Theorem 12.2 ... ...

Theorem 12.2 and its corollaries read as follows:
View attachment 8795
View attachment 8796In the above text from Browder, we read the following:

" ... ... 12.4 Corollary. If \(\displaystyle x \in V\), and \(\displaystyle x \neq 0\), there exists \(\displaystyle \alpha \in V^*\) such that \(\displaystyle \alpha (x) \neq 0\) ... ... "
Can someone please demonstrate a formal and rigorous proof for Corollary 12.4 ...?
Help will be appreciated ... ...

Peter
 

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  • #2
Hi Peter,

Given a nonzero element $\bf{x}$ of $V$, the set $\{\bf{x}\}$ is linearly independent; this set thus extends to a basis of $V$. By the theorem, the basis has a dual basis in $V^*$ in which one element, call it $\alpha$, satisfies $\alpha(\mathbf{x}) = 1$. In particular, $\alpha(\mathbf{x}) \neq 0$.
 
  • #3
Or put it otherwise, let
$$x=\sum_{j=1}^{n} \xi^j u_j \in V$$
in terms of the basis $\{u_1, \cdots, u_n \} $ of $V$, such that $\alpha(x)=0$ for all $\alpha \in V^*$

Then for each member $\bar{u}^i$ of the basis $\{ \bar{u}^1, \cdots, \bar{u}^n\}$ of $V^*$ we have
$$ \bar{u}^i(x)=\xi^i=0$$

Therefore $x=0$

Edit: I changed the positions of some indices.
 
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FAQ: The Dual Space and Covectors .... Browdwer, Theorem 12.2 and Corollaries ....

What is the Dual Space and Covectors?

The dual space and covectors are mathematical concepts used in linear algebra and functional analysis. The dual space is the set of all linear functionals on a vector space, while covectors are elements of the dual space. In simpler terms, the dual space is a space of linear functions that take vectors as inputs and output real numbers, while covectors are the actual linear functions themselves.

What is Theorem 12.2 and its significance?

Theorem 12.2 is a theorem in functional analysis that states that every linear functional on a finite-dimensional vector space can be represented as a linear combination of the dual basis elements. This theorem is significant because it provides a way to represent linear functionals in a more manageable and understandable form, making it easier to solve problems involving them.

How do Corollaries relate to Theorem 12.2?

Corollaries are statements that follow directly from a theorem. In the case of Theorem 12.2, the corollaries are additional statements that can be derived from the theorem and provide further insight and understanding into the properties of dual spaces and covectors.

Can Theorem 12.2 be applied to infinite-dimensional vector spaces?

Yes, Theorem 12.2 can be applied to infinite-dimensional vector spaces, but with some modifications. In this case, the dual space is no longer the set of all linear functionals, but rather the set of all continuous linear functionals. The proof of Theorem 12.2 also requires the use of concepts from functional analysis, such as the Hahn-Banach theorem.

How is Theorem 12.2 used in practical applications?

Theorem 12.2 is used in various areas of mathematics and physics, such as optimization problems, partial differential equations, and quantum mechanics. It provides a framework for understanding linear functionals and their properties, making it a useful tool in solving problems involving them.

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