Variation of the triangle inequality on arbitrary normed spaces

In summary, the triangle inequality is a fundamental concept in mathematics that states the sum of the lengths of any two sides of a triangle must be greater than the length of the remaining side. This applies to arbitrary normed spaces where it can be expressed as ||x + y|| ≤ ||x|| + ||y||, and is significant in defining concepts such as convergence, continuity, and completeness. The triangle inequality also has connections to other mathematical concepts and has various applications in different fields. It cannot be violated in arbitrary normed spaces as it is a fundamental property of these spaces.
  • #1
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The following inequality can easily be proved on ##ℝ## :

## ||x|-|y|| \leq |x-y| ##

I was wondering if it extends to arbitrary normed linear spaces, since I can't seem to prove it using the axioms for linear spaces. (I can however, prove it using the definition of the norm on ##ℝ## by using casework).

Suggestions? Hints?

BiP
 
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  • #2
This is called the reverse triangle inequality and can be shown by applying the regular triangle inequality to

$$ \begin{split} & \| x\| = \| (x - y) + y \|, \\
& \| y\| = \| (y - x) + x \|. \end{split}$$
 
  • #3
The "triangle inequality" itself is part of the definition "norm".
 

1. What is the triangle inequality in mathematics?

The triangle inequality is a fundamental concept in mathematics that states that the sum of the lengths of any two sides of a triangle must be greater than the length of the remaining side. In other words, the shortest distance between two points is a straight line.

2. How does the triangle inequality apply to arbitrary normed spaces?

In arbitrary normed spaces, the triangle inequality can be expressed as ||x + y|| ≤ ||x|| + ||y||, where x and y are elements of the space and ||.|| denotes the norm of the space. This means that the distance between two points in a normed space must be less than or equal to the sum of the distances between each point and a third point.

3. What is the significance of the triangle inequality on arbitrary normed spaces?

The triangle inequality is important in normed spaces because it serves as a measure of distance and allows us to define important concepts such as convergence, continuity, and completeness. It also plays a crucial role in the proof of many theorems in functional analysis and other areas of mathematics.

4. Can the triangle inequality be violated in arbitrary normed spaces?

No, the triangle inequality holds true in all arbitrary normed spaces. If it is violated, then the space does not satisfy the axioms of a normed space and is therefore not considered a valid mathematical structure.

5. How does the triangle inequality relate to other mathematical concepts?

The triangle inequality has connections to other mathematical concepts such as the Cauchy-Schwarz inequality, the reverse triangle inequality, and the parallelogram law. It also has applications in geometry, physics, and computer science, among others.

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