Validity of proof of Cauchy-Schwarz inequality

In summary, the Cauchy-Schwarz inequality is a fundamental mathematical concept that relates the dot product of two vectors to their respective lengths. It can be proven using various methods and has many applications in fields such as linear algebra, geometry, and statistics. It can also be extended to higher dimensions and is always true, making it a powerful tool in mathematical analysis.
  • #1
HaniZaheer
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Proof: If either x or y is zero, then the inequality |x · y| ≤ | x | | y | is trivially correct because both sides are zero.
If neither x nor y is zero, then by x · y = | x | | y | cos θ,
|x · y|=| x | | y | cos θ | ≤ | x | | y |
since -1 ≤ cos θ ≤ 1

How valid is this a proof of the Cauchy-Schwarz inequality?
 
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  • #2
It looks good as long as x · y = | x | | y | cos θ is given by definition or is already proven. (It's good form to indicate "by definition" or "by Lemma xxx", etc.)
 
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Likes HaniZaheer
  • #3
Alright, thanks a lot
 

1. What is the Cauchy-Schwarz inequality?

The Cauchy-Schwarz inequality is a mathematical concept that states that the dot product of two vectors is always less than or equal to the product of their magnitudes. In other words, it shows the relationship between the inner product of two vectors and their respective lengths.

2. How is the Cauchy-Schwarz inequality proven?

The Cauchy-Schwarz inequality can be proven using various methods, including geometric proofs, algebraic proofs, and proofs by induction. One of the most common methods is by using the Cauchy-Schwarz inequality in conjunction with other well-known theorems, such as the triangle inequality and the AM-GM inequality.

3. Why is the Cauchy-Schwarz inequality important?

The Cauchy-Schwarz inequality is a fundamental concept in mathematics and has many applications in different fields, including linear algebra, geometry, and statistics. It is also used in various mathematical proofs and is a building block for other important theorems.

4. Can the Cauchy-Schwarz inequality be extended to higher dimensions?

Yes, the Cauchy-Schwarz inequality can be extended to higher dimensions. In fact, it is a special case of the more general Hölder's inequality, which holds for any number of vectors in any finite-dimensional vector space.

5. Is the Cauchy-Schwarz inequality always true?

Yes, the Cauchy-Schwarz inequality is always true and holds in all vector spaces. This means that it is a universal concept that can be applied to any set of vectors, making it a powerful tool in mathematical analysis and problem-solving.

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