To demonstrate that the unit sphere is an infinite set with respect to the norm, let's consider a specific norm, such as the Euclidean norm. The unit sphere in n-dimensional Euclidean space is defined as the set of all points at a distance of 1 from the origin. Mathematically, it is represented as the set of vectors \( x = (x_1, x_2, ..., x_n) \) such that \( ||x||_2 = \sqrt{x_1^2 + x_2^2 + ... + x_n^2} = 1 \). Frstly, recognize that for any point on the unit sphere, scaling the vector by a positive constant while maintaining its direction still results in a point on the unit sphere. This is due to the homogeneity property of norms. Therefore, for any point \( x \) on the unit sphere, \( cx \) is also on the unit sphere for any \( c \neq 0 \). Now, consider the set of all points \( x = (1, 0, ..., 0) \), \( x = (0, 1, ..., 0) \), and so on, where each point has a 1 in one coordinate and zeros elsewhere. These points form the basis of an infinite set lying on the unit sphere, as scaling them by any positive constant still results in a point on the unit sphere. This demonstrates that the unit sphere is indeed an infinite set concerning the Euclidean norm. In a broader context, this idea can be extended to norms other than the Euclidean norm, emphasizing the generality of the argument. The infinite nature of the unit sphere is a fundamental concept in normed spaces, revealing the richness and complexity of mathematical structures.