# Open Sets in R^n: Show Dot Product Is Open

• seydunas
In summary, the conversation discusses showing the dot product of open sets in R^n is open. It is shown that the composition of sums and products is open, and it suffices to show this for dot product. The dot product is defined as the Cartesian product of open sets.
seydunas
Hi

i want to solve a problem about topology or analysis: Let U and V be open sets in R^n, i want to to show their dot product is open in R.

Tahnk you

Hi seydunas!

The dot product is a composition of sums and products, so it actually suffices to show that sums and products are open. Can you show that?

Hi Micromass,

i have showed that sums and products are open, i.e sum of two open sets and product of two open sets open, in fact sum of an open set and arbitrary set is open, ok, but i can't understand how it suffices to show their composition is also open. Can you explain it more detail.

Let $f:\mathbb{R}\times\mathbb{R}\rightarrow \mathbb{R}$ denote sum and let $g:\mathbb{R}\times\mathbb{R}\rightarrow \mathbb{R}$ denote product. Then, for example in $\mathbb{R}^2$

$$(x,y)\cdot (x^\prime,y^\prime)=xx^\prime+yy^\prime=f(g(x ,x^\prime),g(y,y^\prime))$$

So you see that this dot product is simply the composition of sum and products.

Sorry if I am being thick here, but how do you define the dot product of open sets? Is it the union <a,B> for fixed a in A, and all b in B?

Bacle said:
Sorry if I am being thick here, but how do you define the dot product of open sets? Is it the union <a,B> for fixed a in A, and all b in B?

Yes, I took it as

$$\{<a,b>~\vert~a\in A,b\in B\}$$

I think you mean "Cartesian product", not "dot product".

## 1. What are open sets in R^n?

Open sets in R^n are subsets of the n-dimensional Euclidean space that do not contain their boundary points. In other words, for any point in an open set, there exists a small enough ball around that point that is completely contained within the set.

## 2. How are open sets related to the dot product in R^n?

The dot product is a mathematical operation that takes two vectors and produces a single scalar value. In R^n, open sets are defined using the dot product as a way to measure the distance between points in the space. This allows us to define open sets in a way that is independent of the specific vector space we are working with.

## 3. Why is it important to show that the dot product is open?

Showing that the dot product is open is important because it allows us to prove important theorems about open sets in R^n. For example, the open mapping theorem states that a continuous linear map between two topological vector spaces is either open or closed. This theorem is crucial in many areas of mathematics and physics.

## 4. How can we show that the dot product is open?

We can show that the dot product is open by using the definition of open sets in R^n and the properties of the dot product. Specifically, we can use the Cauchy-Schwarz inequality to show that the dot product is continuous, and then use this fact to show that it is open.

## 5. What are some applications of open sets in R^n?

Open sets in R^n have many applications in mathematics, physics, and engineering. They are essential in the study of topology, functional analysis, and differential equations. In physics, open sets are used to define the phase space of a system, and they play a crucial role in the study of chaos theory and dynamical systems. In engineering, open sets are used to define feasible regions in optimization problems and to approximate solutions to differential equations.

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