# Prove Equivalent Norms: Norm 1 &amp; Norm 2

• Oster
In summary, norms ||.||1 and ||.||2 are equivalent if and only if there exist 2 constants c and k such that c*||x||1 <= ||x||2 <= k*||x||1 for all x in the concerned vector space V. This is equivalent to saying that for every open ball in norm 1, there exists an open ball in norm 2 contained within it and vice versa. This definition is also equivalent to saying that a sequence converges in one norm if and only if it converges in the other, and this follows from the analogous result for equivalent metrics.
Oster
Prove that two norms ||.||1 and ||.||2 are equivalent if and only if there exist 2 constants c and k such that c*||x||1 <= ||x||2 <= k*||x||1 for all x in the concerned vector space V.

Attempt-> Equivalence implies a ball in norm 1 admits a ball in norm 2 and vice versa. For normed linear spaces, I know that B(x,r) = x + r*B(0,1).

So, a ball with respect to norm 1, B1(x,r), admits a ball in norm 2 with say radius 's'.

Using the normed linear space property, I can conclude that for a vector 'y' in V, if ||y||2 < s
then ||y||1 < r.

I don't know where I am going =(

Hi Oster!

What is your definition of equivalent norms?

HI! I got it =D

My definition was that for every open ball with respect to norm 1, there existed an open ball w.r.t norm 2 contained in it and vice versaaaaaaa!

Converse was easy pffff.

That is the same as saying that a sequence converges in one norm if and only if it converges in the other.

In another thread last night you were studying the analogous result for equivalent metrics. This follows from that result, as the norm induces a metric.

## 1. What is the concept of equivalent norms?

The concept of equivalent norms is a mathematical term used to describe two different ways of measuring the "size" of a vector or function. Two norms are considered equivalent if they produce similar results when measuring the size of a vector or function.

## 2. How can we prove that two norms are equivalent?

To prove that two norms, norm 1 and norm 2, are equivalent, we need to show that there exists positive constants c and C such that for any vector or function x, the following inequality holds: c*norm 1(x) ≤ norm 2(x) ≤ C*norm 1(x). This means that the two norms will produce similar results, but one may be multiplied by a constant to get the other.

## 3. What are the implications of having equivalent norms?

Having equivalent norms means that the two ways of measuring the size of a vector or function are essentially the same. This can be useful in various mathematical applications, as it allows us to use whichever norm is more convenient for a given problem.

## 4. Can equivalent norms be proven for any type of vector or function?

Yes, equivalent norms can be proven for any type of vector or function. However, the specific constants c and C may vary depending on the norms being compared and the properties of the vector or function being measured.

## 5. How important is the concept of equivalent norms in mathematics?

The concept of equivalent norms is very important in mathematics, particularly in areas such as functional analysis, optimization, and linear algebra. It allows us to compare different ways of measuring the size of vectors or functions and provides a deeper understanding of mathematical structures and relationships.

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