Gram-Schmidt Orthonormalization .... Garling, Corollary 11.4.2 ....

  • MHB
  • Thread starter Math Amateur
  • Start date
In summary, the conversation revolved around understanding a proof in D. J. H. Garling's book "A Course in Mathematical Analysis: Volume II: Metric and Topological Spaces, Functions of a Vector Variable" in Chapter 11: Metric Spaces and Normed Spaces. The proof in question involved Corollary 11.4.2 and the claim that $\text{span}(e_{k+1}, \ldots, e_d) \subseteq W^\bot$. The conversation ended with Opalg providing a rigorous argument for this claim using Theorem 11.4.1 (Gram-Schmidt Orthonormalization). The original speaker, Peter, expressed gratitude for the help.
  • #1
Math Amateur
Gold Member
MHB
3,998
48
I am reading D. J. H. Garling's book: "A Course in Mathematical Analysis: Volume II: Metric and Topological Spaces, Functions of a Vector Variable" ... ...

I am focused on Chapter 11: Metric Spaces and Normed Spaces ... ...

I need some help to fully understand the proof of Corollary 11.4.2 ...Garling's statement and proof of Corollary 11.4.2 reads as follows:View attachment 8971In the third sentence of the above proof by Garling we read that:

\(\displaystyle \text{span}( e_{k + 1}, \ ... \ ... , e_d ) \subseteq W^{ \bot }\) Although this claim looks very plausible I am unable to formulate a rigorous demonstration of its truth ...

Can someone provide a rigorous argument showing that \(\displaystyle \text{span}( e_{k + 1}, \ ... \ ... , e_d ) \subseteq W^{ \bot }\) ... ... ?
Help will be appreciated ...

Peter
==========================================================================================

Since I suspect a rigorous proof of the above claim of interest will involve Theorem 11.4.1 ( Gram-Schmidt Orthonormalization ... ) ... I am providing text of the same ... as follows:
View attachment 8972
View attachment 8973
Hope that helps ...

Peter
 

Attachments

  • Garling - Corollary 11.4.2 ... .png
    Garling - Corollary 11.4.2 ... .png
    20 KB · Views: 83
  • Garling - 1 - Theorem 11.4.1 ... G_S Orthonormalisation plus Remarks  ... PART 1 .png
    Garling - 1 - Theorem 11.4.1 ... G_S Orthonormalisation plus Remarks ... PART 1 .png
    47.6 KB · Views: 80
  • Garling - 2 - Theorem 11.4.1 ... G_S Orthonormalisation plus Remarks  ... PART 2 ... .png
    Garling - 2 - Theorem 11.4.1 ... G_S Orthonormalisation plus Remarks ... PART 2 ... .png
    7.6 KB · Views: 62
Physics news on Phys.org
  • #2
Peter said:
Can someone provide a rigorous argument showing that \(\displaystyle \text{span}( e_{k + 1}, \ ... \ ... , e_d ) \subseteq W^{ \bot }\) ... ... ?
Peter
If $1\leqslant i\leqslant k$ and $k+1\leqslant j\leqslant d$ then $\langle e_i,e_j\rangle = 0$. So $e_j$ is orthogonal to each member of the orthonormal basis of $W$, and therefore orthogonal to everything in $W$. In other words, $e_j\in W^\bot$. Since that holds for all $j$ with $k+1\leqslant j \leqslant d$, it follows that $\text{span}(e_{k+1},\ldots,e_d) \subseteq W^\bot$.
 
  • #3
Opalg said:
If $1\leqslant i\leqslant k$ and $k+1\leqslant j\leqslant d$ then $\langle e_i,e_j\rangle = 0$. So $e_j$ is orthogonal to each member of the orthonormal basis of $W$, and therefore orthogonal to everything in $W$. In other words, $e_j\in W^\bot$. Since that holds for all $j$ with $k+1\leqslant j \leqslant d$, it follows that $\text{span}(e_{k+1},\ldots,e_d) \subseteq W^\bot$.
Thanks for the help, Opalg ...

Much appreciated!

Peter
 

FAQ: Gram-Schmidt Orthonormalization .... Garling, Corollary 11.4.2 ....

1. What is Gram-Schmidt Orthonormalization?

Gram-Schmidt Orthonormalization is a mathematical process used to transform a set of linearly independent vectors into a set of orthonormal vectors. This process is commonly used in linear algebra and functional analysis to simplify calculations and solve problems involving vector spaces.

2. How does Gram-Schmidt Orthonormalization work?

The Gram-Schmidt Orthonormalization process involves taking a set of linearly independent vectors and systematically transforming them into a set of orthonormal vectors. This is done by first normalizing the first vector, then subtracting its projection onto the second vector from the second vector, and so on for each subsequent vector in the set.

3. What is the purpose of Gram-Schmidt Orthonormalization?

The purpose of Gram-Schmidt Orthonormalization is to simplify calculations involving vector spaces by transforming a set of linearly independent vectors into a set of orthonormal vectors. Orthonormal vectors have a magnitude of 1 and are perpendicular to each other, making them easier to work with in mathematical calculations.

4. Can Gram-Schmidt Orthonormalization be used for any set of vectors?

Yes, Gram-Schmidt Orthonormalization can be used for any set of linearly independent vectors. However, it is important to note that the resulting orthonormal vectors may not be unique and may depend on the order in which the vectors are processed.

5. What is Corollary 11.4.2 in Garling's book about Gram-Schmidt Orthonormalization?

Corollary 11.4.2 in Garling's book is a mathematical statement that states that if a set of vectors is linearly independent, then the Gram-Schmidt Orthonormalization process will produce a set of orthonormal vectors with the same span as the original set. This corollary is useful for proving the existence of orthonormal bases for vector spaces.

Back
Top