# Understanding Schwarz Inequality and Its Role in Higher Dimensions

• ahmed markhoos
In summary, the Schwarz inequality states that the absolute value of the dot product of two vectors is less than or equal to the product of their magnitudes. This is true if and only if the cosine of the angle between the vectors is less than or equal to 1. The inequality holds for all dimensions and the angle between two vectors is defined using the regular cosine function. There is nothing special about higher dimensions that allows the cosine to be greater than 1.
ahmed markhoos
Hello,

I'm having a small problem with Schwarz inequality, |u⋅v|≤||u||||v||

the statement is true if and only if cosΘ≤1 !, I'm familiar with this result but how could it be more than 1?
what is so special in higher dimensions that it gave the ability for cosine to be more than 1? why and how?

It can't be more than 1. The inequality ##|\langle u,v\rangle|\leq\|u\|\|v\|## holds for all vectors u and v. That's why it makes sense to define "the angle between u and v" as the ##\theta## such that
$$\cos\theta =\frac{\langle u,v\rangle}{\|u\|\|v\|}.$$ This isn't some new version of the cosine function. It's the plain old cosine function that you're familiar with.

Last edited:
manoj@kumar and ahmed markhoos

## 1. What is Schwarz inequality?

Schwarz inequality, also known as Cauchy-Schwarz inequality, is a fundamental theorem in mathematics that relates the inner product of two vectors to their norms. It states that the absolute value of the inner product of two vectors is less than or equal to the product of their norms.

## 2. How is Schwarz inequality used in higher dimensions?

In higher dimensions, Schwarz inequality is used to find bounds on the inner product of vectors in a vector space. This is important in many areas of mathematics, such as linear algebra, functional analysis, and geometry.

## 3. What is the significance of Schwarz inequality in mathematics?

Schwarz inequality is significant because it provides a tool for proving many important theorems in mathematics. It is also a key concept in understanding the relationship between vectors and their norms.

## 4. Can Schwarz inequality be extended to infinite-dimensional spaces?

Yes, Schwarz inequality can be extended to infinite-dimensional spaces. In fact, it is a fundamental theorem in functional analysis and is used to define the norm of a vector in an infinite-dimensional space.

## 5. How is Schwarz inequality related to other mathematical concepts?

Schwarz inequality is closely related to other concepts such as the triangle inequality, the Cauchy-Schwarz master inequality, and the Hölder inequality. It also has applications in other areas of mathematics, such as optimization and probability theory.

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