Question

Let \((X,\|\cdot\|)\) be a Banach space. Show that \(\mu_{B_{X}}(x)=\|x\|\). Hint: Use continuity of the norm.

Short answer

\[\rho(x, B_X) = orm{x}\]

Step-by-step solution

Define the Minkowski functional

To show that \(orm{x}\) is equal to the Minkowski functional \(orm{x}_{B_X}\) over the unit ball \(B_X\), start by understanding the definition of the Minkowski functional for a set \(\rho(x,A)\). For a given set \(A\) and point \(x\), it is defined by: \[\rho(x,A) = \text{inf}\{\lambda > 0 : x \in \lambda A\}\]In our case, \(A = B_X\), the closed unit ball of \((X, orm{\cdot\})\), so we have: \[\rho(x,B_X) = \text{inf}\{\lambda > 0 : x \in \lambda B_X\}\]

Relate the unit ball to the norm

Since \(B_X = \{y \in X : orm{y} \le 1\}\), for \(x \in X\), we require \(x \in \lambda B_X\). This implies \(orm{\frac{x}{\lambda}} \le 1\), or equivalently, \(orm{x} \le \lambda\). Therefore, \(\lambda\ge orm{x}\).

Infimum property

Using the infimum property of the Minkowski functional, we have: \[\rho(x, B_X) = \text{inf}\{\lambda > 0 : orm{x} \le \lambda\} .\] Given that \(\orm{x}\) is the smallest possible \(\lambda\) that satisfies this inequality, it follows that \(\rho(x, B_X) = orm{x}\).

Use continuity of the norm

Because the norm is continuous in Banach spaces, the equality extends to every \(x \in X\). Hence, by the continuity of \(\orm{\cdot}\) and the properties shown: \[\rho(x, B_X) = orm{x}\].

Key concepts

Banach space

A Banach space is a complete normed vector space. This means every Cauchy sequence (a sequence where elements get arbitrarily close to each other) in the space converges to a limit within the space.
Banach spaces are significant in various areas of analysis and have applications in functional analysis and PDEs. They allow for the analysis of spaces of functions and operators. Key properties include:

• Norm: A function that assigns a non-negative length or size to all vectors in a vector space.
• Completeness: Every Cauchy sequence in the space converges to a limit within the space.

Minkowski functional

The Minkowski functional, also known as the gauge function, is a useful concept in Banach spaces. For a point \(x\) and a set \(A\), it is defined as:
\[ \rho(x, A) = \inf \{ \lambda > 0 : x \in \lambda A \} \] It measures the 'scaling' needed for a point \(x\) to belong to a set \(A\). Specifically, it's often applied to the unit ball in Banach spaces. This helps in linking the norm of a vector to its behavior in relation to a set.

Norm continuity

In Banach spaces, the norm function \( \| \cdot \| \) is continuous. This means small changes in the elements of the space result in small changes in their norm. Continuity of norm is crucial for understanding how elements transform and behave under limits. Formally, we say the norm is continuous if for every sequence \( \{x_n\} \subset X \) converging to \( x \in X \), we have \( \| x_n \| \to \| x \| \).

Unit ball

The unit ball in a Banach space, denoted \( B_X \), is the set of all vectors whose norm is less than or equal to one:
\[ B_X = \{ y \in X : \| y \| \le 1 \} \] The unit ball helps visualize and manipulate elements of the space. It's critical in defining and understanding the Minkowski functional, among other applications. For example, if we need \(x \in \lambda B_X \), it implies \( \| \frac{x}{\lambda} \| \le 1 \).

Infimum property

The infimum property is about finding the greatest lower bound of a set. In the context of the Minkowski functional and Banach spaces, it means finding the smallest scalar \( \lambda \) so that \(x\) fits within \(\lambda A\). Formally, if we have:
\[ \rho(x, B_X) = \inf \{ \lambda > 0 : \| x \| \le \lambda \} \] Here, the smallest \( \lambda \) satisfying the inequality is \( \| x \| \). Thus, \( \rho(x, B_X) = \| x \| \). This property underscores the tight relationship between the norm and the Minkowski functional.