# R^n as a normed space .... D&K Lemma 1.1.7 .... .... some inequalities ....

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In summary, Duistermaat and Kolk's Lemma 1.1.7 says that if x is a vector and y is a vector such that y > x, then x^2 + y^2 is greater than x^2.
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I am reading "Multidimensional Real Analysis I: Differentiation by J. J. Duistermaat and J. A. C. Kolk ...

I am focused on Chapter 1: Continuity ... ...

I need help with an aspect of Lemma 1,1,7 (iv) ...

Duistermaat and Kolk"s Lemma 1.1.7 reads as follows:
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At the start of the proof of (iv) we read the following:

" ... ... $$\displaystyle \mid x_j \mid \le \left( \sum_{ 1 \le j \le n } \mid x_j \mid^2 \right)^{ \frac{ 1 }{ 2 } } = \| x \|$$ ... ... ... "
Suppose now we want to show, formally and rigorously that $$\displaystyle \mid x_j \mid \le \left( \sum_{ 1 \le j \le n } \mid x_j \mid^2 \right)^{ \frac{ 1 }{ 2 } }$$Maybe we could start with (obviously true ...)

$$\displaystyle \mid x_j \mid = ( \mid x_j \mid^2 )^{ \frac{ 1 }{ 2 } }$$ ... ... ... ... ... (1)

then we can write

$$\displaystyle \mid x_j \mid = ( \mid x_j \mid^2 )^{ \frac{ 1 }{ 2 }} \le ( \mid x_1 \mid^2 + \mid x_2 \mid^2 + \ ... \ ... \ + \mid x_j \mid^2 + \ ... \ ... \ + \mid x_n \mid^2 )^{ \frac{ 1 }{ 2 } }$$and we note that$$\displaystyle ( \mid x_1 \mid^2 + \mid x_2 \mid^2 + \ ... \ ... \ + \mid x_j \mid^2 + \ ... \ ... \ + \mid x_n \mid^2 )^{ \frac{ 1 }{ 2 } } = ( x_1^2 + x_2^2 + \ ... \ ... \ + x_j^2 + \ ... \ ... \ + x_n^2 )^{ \frac{ 1 }{ 2 } } = \| x \|$$ ... ... ... ... (3) ... BUT ... I worry that (formally anyway) (1) is invalid ... or compromised at least ...

... for suppose for example $$\displaystyle x_j = -3$$ then ...... we have LHS of (1) = $$\displaystyle \mid x_j \mid = \mid -3 \mid = 3$$... BUT ...

RHS of (1)$$\displaystyle = ( \mid x_j \mid^2 )^{ \frac{ 1 }{ 2 } } = ( \mid -3 \mid^2 )^{ \frac{ 1 }{ 2 } } = ( 3^2 )^{ \frac{ 1 }{ 2 } } = 9^{ \frac{ 1 }{ 2 } } = \pm 3$$My question is as follows:

How do we deal with the above situation ... and

... how do we formally and rigorously demonstrate that $$\displaystyle \mid x_j \mid \le \left( \sum_{ 1 \le j \le n } \mid x_j \mid^2 \right)^{ \frac{ 1 }{ 2 } }$$Hope someone can help ...

Peter

Last edited:
RHS of (1)$$\displaystyle = ( \mid x_j \mid^2 )^{ \frac{ 1 }{ 2 } } = ( \mid -3 \mid^2 )^{ \frac{ 1 }{ 2 } } = ( 3^2 )^{ \frac{ 1 }{ 2 } } = 9^{ \frac{ 1 }{ 2 } } = \pm 3$$

How do we deal with the above situation ... and

$9^{1/2}\ne\pm 3\;rather\;9^{1/2} = 3$

You may be thinking about solutions to $x^2 = 9$. That's a different animal.
It is appropriate to say $\sqrt{x^{2}} = |x|$ but only if we know nothing of x.

... how do we formally and rigorously demonstrate that $$\displaystyle \mid x_j \mid \le \left( \sum_{ 1 \le j \le n } \mid x_j \mid^2 \right)^{ \frac{ 1 }{ 2 } }$$

(x + y)^2 = x^2 + 2xy + y^2 = x^2 + y^2 + Stuff

(x + y + z)^2 = x^2 + y^2 + z^2 + Stuff

Now, we need to show only that Stuff >= 0.

Just an idea. Let's see where you go with it.

tkhunny said:
$9^{1/2}\ne\pm 3\;rather\;9^{1/2} = 3$

You may be thinking about solutions to $x^2 = 9$. That's a different animal.
It is appropriate to say $\sqrt{x^{2}} = |x|$ but only if we know nothing of x.
(x + y)^2 = x^2 + 2xy + y^2 = x^2 + y^2 + Stuff

(x + y + z)^2 = x^2 + y^2 + z^2 + Stuff

Now, we need to show only that Stuff >= 0.

Just an idea. Let's see where you go with it.
Hi tkhunny,

Thanks for the help ...

You're right of course ...

Thanks again ...

Peter

## 1. What does the notation "R^n" mean in this context?

The notation "R^n" refers to the n-dimensional Euclidean space, which is a mathematical concept that represents n-tuples of real numbers. In this context, it is being used to define a normed space, which is a mathematical structure that consists of a set of elements and a function that assigns a non-negative "length" or "size" to each element.

## 2. What is a normed space?

A normed space is a mathematical structure that consists of a set of elements and a norm function that assigns a non-negative "length" or "size" to each element. Normed spaces are used in functional analysis and other areas of mathematics to study the properties of vector spaces and their elements.

## 3. What is Lemma 1.1.7 in the D&K textbook?

Lemma 1.1.7 is a specific lemma (a type of mathematical proof) found in the D&K textbook, which is a commonly used textbook in the field of functional analysis. This particular lemma is used to prove certain inequalities related to normed spaces, which are mathematical structures that consist of a set of elements and a function that assigns a non-negative "length" or "size" to each element.

## 4. What are some examples of inequalities that may be proved using Lemma 1.1.7?

Lemma 1.1.7 can be used to prove a variety of inequalities related to normed spaces, such as the triangle inequality (which states that the length of the sum of two vectors is always less than or equal to the sum of their individual lengths) or the Cauchy-Schwarz inequality (which states that the inner product of two vectors is always less than or equal to the product of their individual norms).

## 5. How is R^n as a normed space relevant in scientific research?

R^n as a normed space is relevant in scientific research because it provides a mathematical framework for studying and analyzing various phenomena. For example, in physics, normed spaces are used to represent vector spaces and their elements, which are essential in understanding and predicting the behavior of physical systems. In engineering, normed spaces are used to model and analyze complex systems, such as transportation networks or communication systems. Overall, R^n as a normed space is a fundamental concept that is utilized in many different fields of scientific research.

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