\[ \newcommand{R}{\mathbb{R}} \newcommand{defarrow}{\quad \stackrel{\text{def}}{\Longleftrightarrow} \quad} \]
Balls and spheres
Let \((X,d)\) be a metric space with distance \(d\colon X \times X \to [0,\infty)\).
Two important concepts are the ball of radius \(r > 0\) centered at \(x_0 \in X\),
\[
B_r(x_0) := \{ x \in X \colon d(x,x_0) < r \},
\]
and the sphere of radius \(r > 0\) centered at \(x_0 \in X\),
\[
S_r(x_0) := \{ x \in X \colon d(x,x_0) = r \}.
\]
For normed spaces, or other vector spaces that are also metric spaces, we simply write
\[
B_r := B_r(0)\quad \text{ and }\quad S_r = S_r(0),
\]
for balls and spheres centered at the origin (zero element). The sets \(B_1\) and \(S_1\) are called the unit ball and unit sphere, respectively.
- The ball of radius \(2\) centered at \( (1,0) \) in Euclidean space \(\R^2\):
\[ B_2((1,0)) = \{ (x,y) \in \R^2 \colon (x-1)^2 + y^2 < 4\}. \]
- Sequence spaces are spaces in which each element \[ x = \{x_n\}_{n \in \mathbb N} = (x_1,x_2, \ldots)\] is a sequence (usually of real or complex numbers). The most important ones are the so-called \(l_p\)-spaces:
- Let \(l_\infty\) be the space of sequences \(\{x_j\}_{j\in\mathbb N} = (x_1, x_2, \ldots)\) for which \[ \|x\|_{l_\infty} = \sup_{j \in \mathbb N} |x_j| < \infty.\] Then the sequence \((1/2,2/3,3/4, \ldots) \in S_1\) in \(l_\infty\) (since \( \sup_{j \in \mathbb N}| \frac{j}{j+1} | = 1\)).
- For any \(p \geq 1\), let \(l_p\) be the space of sequences \(\{x_j\}_{j\in\mathbb N} = (x_1, x_2, \ldots)\) for which \[\|x\|_{l_p} = \big( \sum_{j \in \mathbb N} |x_j|^p \big)^{1/p}< \infty.\] Let further \[e_1 = (1,0,0,\ldots), \quad e_2 = (0,1,0,0\ldots) \quad\text{ and }\quad e_j = (\ldots,0,1,0,\ldots), \quad j \in \mathbb N.\] Then \[ e_j \in S_1 \quad\text{ for all }\quad j \in \mathbb N,\] but \[ d(e_i,e_j) = \| e_i - e_j \|_{l_p} = (|1|^p + |-1|^p)^{1/p} = 2^{1/p} \geq 1 \quad\text{ whenever }\quad i \neq j.\] Note that such a sequence of elements could never exist in \(\mathbb R^n\) (or any other finite-dimensional vector space).
- The unit ball in \(BC([0,1],\mathbb R)\) consists of all functions whose graph \(y = f(x)\) lies strictly between the lines \(y=\pm 1\).
N.b. The unit ball may look quite different depending on the underlying metric/norm. The following illustration captures this in the case of the \(l_p\)-norm on \(\R^2\). Homogeneity and the triangle inequality however imply that a ball in any metric given by a norm will always be a convex set in the underlying space.
