Justifying Set Boundedness of $S_{||\cdot||_2}$ in $\mathbb{R}^n

In summary, set boundedness in $\mathbb{R}^n$ is justified by showing that all elements in the set have a finite distance from the origin, which can be achieved by using the $\| \cdot \|_2$ norm. This norm measures the Euclidean distance between a point and the origin in $\mathbb{R}^n$, and if the norm of each point in the set is less than or equal to some finite value, then the set is considered bounded. Other norms, such as the $\| \cdot \|_1$ norm and the $\| \cdot \|_\infty$ norm, can also be used to justify set boundedness. This concept is important in mathematics and science because it
  • #1
evinda
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Hello! (Wave)We have that $S_{||\cdot||_2}:= \{ x \in \mathbb{R}^n: ||x||_2=1\}$.

How can we justify that the above set is bounded?

Do we just say that if $x \in S_{||\cdot||_2}$ then $||x||_2=1 \leq 1$ and so the set is bounded. How could we justify it more formally?
 
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  • #2
Hi evinda,

To show formally that this set is bounded, you need to prove that there is a positive number $K$ such that for all $x,y\in S$, $\|x-y\|_2\le K$. Using the triangle inequality you'll find that $K=2$ is a suitable upper bound.
 

FAQ: Justifying Set Boundedness of $S_{||\cdot||_2}$ in $\mathbb{R}^n

How is set boundedness justified in $\mathbb{R}^n$?

Set boundedness in $\mathbb{R}^n$ is justified by showing that all elements in the set have a finite distance from the origin. This can be achieved by using the $\| \cdot \|_2$ norm, which measures the Euclidean distance between a point and the origin in $\mathbb{R}^n$. If the norm of each point in the set is less than or equal to some finite value, then the set is considered bounded.

What is the $\| \cdot \|_2$ norm and how does it relate to set boundedness?

The $\| \cdot \|_2$ norm, also known as the Euclidean norm, is a function that calculates the distance between a point and the origin in $\mathbb{R}^n$. In set boundedness, if the norm of each point in the set is less than or equal to some finite value, then the set is considered bounded.

Can set boundedness be justified using other norms besides $\| \cdot \|_2$?

Yes, set boundedness can be justified using other norms such as the $\| \cdot \|_1$ norm or the $\| \cdot \|_\infty$ norm. These norms measure the distance between a point and the origin in different ways, but the concept of boundedness remains the same. A set is considered bounded if the norm of each point is less than or equal to some finite value.

Why is set boundedness important in mathematics and science?

Set boundedness is important in mathematics and science because it allows us to define and analyze sets in a rigorous and meaningful way. Bounded sets have well-defined boundaries, making them easier to work with in mathematical and scientific calculations. Additionally, many important theorems and concepts in mathematics and science rely on the concept of set boundedness.

Are there any real-world applications of set boundedness in $\mathbb{R}^n$?

Yes, set boundedness in $\mathbb{R}^n$ has many real-world applications. For example, in physics, the motion of objects in space can be described using bounded sets. In computer science, algorithms for optimizing functions often involve bounded sets. In engineering, bounded sets are used to define safe operating ranges for systems. Overall, set boundedness is a fundamental concept that has wide-ranging applications in various fields.

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