Question

Let \(\|\cdot\|_{1}\) and \(\|\cdot\|_{2}\) be two equivalent norms on a vector space \(X\). Let \(B_{1}\) and \(B_{2}\) be the closed unit balls of \(\left(X,\|\cdot\|_{1}\right)\) and \(\left(X,\|\cdot\|_{2}\right)\), respectively. Show that \(B_{1}\) and \(B_{2}\) are homeomorphic. Recall that two topological spaces \(K\) and \(L\) are called homeomorphic if there exists a bijection \(\varphi\) from \(K\) onto \(L\) such that \(\varphi\) and \(\varphi^{-1}\) are continuous. Such a \(\varphi\) is called a homeomorphism. Hint: Define a map \(\phi\) from \(B_{1}\) onto \(B_{2}\) by \(\phi(0)=0\) and \(\phi(x)=\frac{\|x\|_{1}}{\|x\|_{2}} x\) for \(x \in B_{1} \backslash\\{0\\} .\) Clearly, \(\|\phi(x)\|_{2}=\|x\|_{1}\), and continuity at 0 follows from the equivalence of the norms.

Short answer

Define \(\backslashphi\) as given. \(\backslashphi\) is bijective and both \(\backslashphi\) and \(\backslashphi^{-1}\) are continuous. Hence, \(\backslashB_{1}\) and \(\backslashB_{2}\) are homeomorphic.

Step-by-step solution

Define the Map

Define a map \(\backslashphi\) from \(\backslashB_{1}\) onto \(\backslashB_{2}\) such that \(\backslashphi(0)=0\) and \(\backslashphi(x)=\backslashfrac{\backslash|x\backslash|_1}{\backslash|x\backslash|_2} x\) for \(x \backslashin \backslashB_{1}\backslashbackslash\backslash{0\backslash}\). This map will help us establish the homeomorphism between the unit balls.

Show \(\backslash|\backslashphi(x)\backslash|_{2} = \backslash| x \backslash|_{1}\)

For \(x \backslashin \backslashB_{1}\backslashbackslash\backslash{0\backslash}\), compute \(\backslash|\backslashphi(x)\backslash|_{2} = \backslash|\backslashfrac{\backslash|x\backslash|_{1}}{\backslash|x\backslash|_{2}} x\|_{2}.\) Simplify this using the properties of scalar multiplication: \( = \backslashfrac{\backslash|x\backslash|_{1}}{\backslash|x\backslash|_{2}} \backslash| x \backslash|_{2} = \backslash| x \backslash|_{1}.\)

Establish that \phi\) is bijective

Argue that \(\backslashphi\) is a bijection by showing that it maps \(\backslashphi : \backslashB_{1} \rightarrow \backslashB_{2}\) surjectively and injectively. Since the norms are equivalent, every \(x \backslashin \backslashB_{1}\) maps uniquely to an element in \(\backslashphi(x) \backslashin \backslashB_{2}\).

Prove Continuity of \(\backslashphi\)

Use the equivalence of the norms to show that as \(x \rightarrow 0\) in the norm \(\backslash| \backslashcdot \backslash|_{1}, \backslash| x \backslash|_{1} \backslashrightarrow 0\). Consequently, \(\backslashphi(x) = \backslashfrac{\backslash|x\backslash|_{1}}{\backslash|x\backslash|_{2}} x\) tends to 0, proving that \(\backslashphi\) is continuous at 0. For other points, \(\backslashphi\)'s definition ensures continuity follows from linearity.

Prove Continuity of the Inverse Map \(\backslashphi^{-1}\)

Show that the inverse map \((\backslashphi^{-1} : \backslashB_{2} \rightarrow \backslashB_{1})\) given by \((\backslashphi^{-1}(y) = \backslashfrac{\backslash|y\backslash|_{2}}{\backslash|y\backslash|_{1}} y\)) is also continuous. Again, this follows from the equivalence of the norms and the similar structure as \(\backslashphi\).

Key concepts

homeomorphism

A homeomorphism is a fundamental concept in topology. It is a mapping between two topological spaces that is a both continuous and bijective, and its inverse is also continuous. This implies that not only is the space itself transformed gracefully from one form to another, but the overall structure of the space is preserved.
To understand this better, imagine that you have a rubber sheet that you can stretch and flex, but not tear or glue. If you can transform one shape into another using only these kinds of changes, then the two shapes are homeomorphic. They are essentially the same from a topological perspective, retaining their intrinsic properties despite their apparent differences.
Here's how it relates to our exercise: If we have two equivalent norms on a vector space, the closed unit balls under these norms can be seen as topologically the same, or homeomorphic, under a specific mapping.

continuous mapping

Continuity is a key concept in both calculus and topology, and it's essential for understanding homeomorphisms.
A function or mapping is continuous if, intuitively speaking, small changes in the input lead to small changes in the output. Formally, a function \(\backslasphi:X\rightarrow Y\) is continuous if for every open set \(U\) in \(Y\), the preimage \(\backslasphi^{-1}(U)\) is an open set in \(X\).
For our exercise, to show that the unit balls under two norms are homeomorphic, we need to establish a mapping between them that is continuous in both directions. We define \(\backslasphi(x)=\backslashfrac{\backslashlvert x\backslashlvert_{1}}{\backslashlvert x\backslashlvert_{2}} x\) for all \(\backslasphi e 0\) and \(\backslasphi(0) = 0\). We need to show this mapping is continuous and that its inverse is also continuous, which follows from the equivalence of the norms.

vector spaces

Vector spaces are a foundational concept in linear algebra and are critical in many areas of mathematics and applied sciences. A vector space, also called a linear space, is a collection of vectors where vector addition and scalar multiplication are defined and satisfy certain axioms like associativity, commutativity, and distributivity.
Some common examples of vector spaces include the set of all vectors in Euclidean space, matrices, polynomials, and functions. Each kind of vector space has different applications, but they all share the basic properties that make them vector spaces.
In our exercise, we are working with a vector space \(X\) on which two norms \(\backslashlvert \backslashcdot\backslashlvert_{1}\) and \(\backslashlvert \backslashcdot\backslashlvert_{2}\) are defined. The norms being equivalent means that they preserve the structure of the vector space in a comparable way. This equivalence allows us to map the unit balls under these norms to each other in a manner that is consistent with the vector space structure, eventually establishing the homeomorphism between the unit balls.