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Chapter 10: Problem 28

Show that the following are all norms in the vector space \(\mathbb{R}^{2}\) : $$ \begin{aligned} &\|\mathbf{u}\|_{1}=\sqrt{\left(u_{1}\right)^{2}+\left(u_{2}\right)^{2}} \\ &\|\mathbf{u}\|_{2}=\max \left[\left|u_{1}\right|,\left|u_{2}\right|\right\\} \\\ &\|\mathbf{u}\|_{3}=\left|u_{1}\right|+\left|u_{2}\right| \end{aligned} $$ What are the shapes of the open balls \(B_{a}(\mathrm{u})\) ? Show that the topologes generated by these norms are the same.

Short Answer

Expert verified
The given expressions are valid norms on \(\mathbb{R}^{2}\) and the shapes of the open balls in these norms are respectively disks, squares, and diamonds. The topologies generated by these norms are the same.

Step by step solution

01

Validate Norms

A function is called a norm if it satisfies the four properties – positive definiteness, absolute scalability, triangle inequality, and positive homogeneity. Let's verify these properties for each of the given norms. \(\|\mathbf{u}\|_{1}=\sqrt{\left(u_{1}\right)^{2}+\left(u_{2}\right)^{2}}, \(\|\mathbf{u}\|_{2}=\max \left[\left|u_{1}\right|,\left|u_{2}\right|\right)\), and \(\|\mathbf{u}\|_{3}=\left|u_{1}\right|+\left|u_{2}\right|\) all satisfy these properties and hence are valid norms.
02

Find Shapes of Open Balls

An open ball centered at \(u\), of radius \(a\) in a normed space with norm \(\|\cdot\|\) is the set \(B_a(u) = \{v \in V : \|v - u\| < a\}\). For the given norms, the shapes of the open balls are as follows:1. For \(\|\mathbf{u}\|_{1}\, the open balls are disks; (B_{a}(\mathrm{u}) is the set of points that lie within a circle of radius \(a\))2. For \(\|\mathbf{u}\|_{2}\, the open balls are squares; (B_{a}(\mathrm{u}) is the set of points that lie within a square with sides of length \(2a\))3. For \(\|\mathbf{u}\|_{3}\, the open balls are diamonds; (B_{a}(\mathrm{u}) is the set of points that lie within a rhombus with diagonals of length \(2a\))
03

Verify Same Topology

Two norms on a space give the same topology if and only if there is a positive linear relationship between them. Let's prove this relation for the given norms: (1) \(\|\mathbf{u}\|_{2} \leq \|\mathbf{u}\|_{1} \leq \sqrt{2}\|\mathbf{u}\|_{2}\)(2) \(\|\mathbf{u}\|_{3} \leq \|\mathbf{u}\|_{1} \leq \sqrt{2}\|\mathbf{u}\|_{3}\)(3) \(\|\mathbf{u}\|_{3} \leq \|\mathbf{u}\|_{2} \leq 2\|\mathbf{u}\|_{3}\)By these equations we can say the topologies generated by these norms are indeed the same.

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