# Parallelogram law calculation is this an error in the text?

• AxiomOfChoice
In summary, the authors of Kolmogorov and Fomin's Real Analysis book state that for any vectors f, g, h in a real Hilbert space, the equation \|f + g + h\|^2 + \|f - h - g\|^2 = 2\|f - h\|^2 + 2\|g\|^2 holds true. They provide an explanation using the parallelogram law, which is valid in any inner product space. However, there is a discrepancy in their claim as it fails in \mathbb R with f = 2, g = -1, h = 1 when the inner product is just multiplication. A possible explanation could be a typo in the equation
AxiomOfChoice
In Kolmogorov and Fomin's Real Analysis book, pg. 161, they make the following claim: For any vectors f,g,h in a real Hilbert space, we have

$$\|f + g + h\|^2 + \|f - h - g\|^2 = 2\|f - h\|^2 + 2\|g\|^2.$$

They attempt to justify this using the parallelogram law:

$$\|x + y\|^2 + \|x - y\|^2 = 2\|x\|^2 + 2\|y\|^2,$$

which holds in any inner product space. But I do not think they're right about this; doesn't their claim fail in $\mathbb R$ with f = 2, g = -1, h = 1, when the inner product is just multiplication? Don't you get something like 8 = 4?

If the first h in your equation had a negative in front of it you would get the result using x=f-h and y=g. Maybe it is just a typo.

For anyone who should happen across this page in the future: I've discovered that this actually is a typo. Click http://math.gmu.edu/~tlim/errataByEdgar.pdf" for more information. Apparently, that entire section of the book (examining when a norm is derived from an inner product) is littered with errors.

Last edited by a moderator:

## 1. What is the Parallelogram law calculation?

The Parallelogram law calculation is a mathematical formula used to find the magnitude and direction of a resultant vector when two vectors are added together.

## 2. How is the Parallelogram law calculation used in science?

The Parallelogram law calculation is commonly used in physics, engineering, and other scientific fields to analyze and understand the motion and forces acting on objects.

## 3. What is the error in the text regarding Parallelogram law calculation?

Without further context, it is impossible to determine what error may be present in the text regarding Parallelogram law calculation. It could be a typo, incorrect formula, or a misinterpretation of the concept.

## 4. Can you provide an example of a Parallelogram law calculation?

Sure, for example, if two vectors A and B are added together to form a resultant vector R, the Parallelogram law states that the magnitude of R will be equal to the diagonal of the parallelogram formed by A and B, and the direction of R will be the angle between A and B.

## 5. How important is it to understand Parallelogram law calculation in science?

It is essential to have a good understanding of Parallelogram law calculation in science as it is a fundamental concept used in many fields of study, and it allows for the accurate analysis of complex systems and phenomena.

• Calculus
Replies
4
Views
1K
• Calculus
Replies
3
Views
1K
• Calculus
Replies
5
Views
868
• Topology and Analysis
Replies
1
Views
752
• Calculus
Replies
9
Views
2K
• Calculus
Replies
1
Views
1K
• Calculus
Replies
4
Views
2K
• Calculus
Replies
18
Views
2K
• General Math
Replies
1
Views
1K
• Differential Equations
Replies
65
Views
3K