stylistic:style_transfer
Equivalence

The concept of unit circle (the set of all vectors of norm 1) is different in different norms: for the 1-norm, the unit circle is a square oriented as a diamond; for the 2-norm (Euclidean norm), it is the well-known unit circle; while for the infinity norm, it is an axis-aligned square. For any p {\displaystyle p} -norm, it is a superellipse with congruent axes (see the accompanying illustration). Due to the definition of the norm, the unit circle must be convex and centrally symmetric (therefore, for example, the unit ball may be a rectangle but cannot be a triangle, and p ≥ 1 {\displaystyle p\geq 1} for a p {\displaystyle p} -norm).

In terms of the vector space, the seminorm defines a topology on the space, and this is a Hausdorff topology precisely when the seminorm can distinguish between distinct vectors, which is again equivalent to the seminorm being a norm. The topology thus defined (by either a norm or a seminorm) can be understood either in terms of sequences or open sets. A sequence of vectors { v n } {\displaystyle \{v_{n}\}} is said to converge in norm to v, {\displaystyle v,} if ‖ v n − v ‖ → 0 {\displaystyle \left\|v_{n}-v\right\|\to 0} as n → ∞. {\displaystyle n\to \infty.} Equivalently, the topology consists of all sets that can be represented as a union of open balls. If (X, ‖ ⋅ ‖) {\displaystyle (X,\|\cdot \|)} is a normed space then ‖ x − y ‖ = ‖ x − z ‖ + ‖ z − y ‖ for all x, y ∈ X and z ∈. {\displaystyle \|x-y\|=\|x-z\|+\|z-y\|{\text{ for all }}x,y\in X{\text{ and }}z\in.}

Two norms ‖ ⋅ ‖ α {\displaystyle \|\cdot \|_{\alpha }} and ‖ ⋅ ‖ β {\displaystyle \|\cdot \|_{\beta }} on a vector space X {\displaystyle X} are called equivalent if they induce the same topology, which happens if and only if there exist positive real numbers C {\displaystyle C} and D {\displaystyle D} such that for all x ∈ X {\displaystyle x\in X} C ‖ x ‖ α ≤ ‖ x ‖ β ≤ D ‖ x ‖ α. {\displaystyle C\|x\|_{\alpha }\leq \|x\|_{\beta }\leq D\|x\|_{\alpha }.} For instance, if p > r ≥ 1 {\displaystyle p>r\geq 1} on C n, {\displaystyle \mathbb {C} ^{n},} then ‖ x ‖ p ≤ ‖ x ‖ r ≤ n (1 / r − 1 / p) ‖ x ‖ p. {\displaystyle \|x\|_{p}\leq \|x\|_{r}\leq n^{(1/r-1/p)}\|x\|_{p}.}

In particular, ‖ x ‖ 2 ≤ ‖ x ‖ 1 ≤ n ‖ x ‖ 2 {\displaystyle \|x\|_{2}\leq \|x\|_{1}\leq {\sqrt {n}}\|x\|_{2}} ‖ x ‖ ∞ ≤ ‖ x ‖ 2 ≤ n ‖ x ‖ ∞ {\displaystyle \|x\|_{\infty }\leq \|x\|_{2}\leq {\sqrt {n}}\|x\|_{\infty }} ‖ x ‖ ∞ ≤ ‖ x ‖ 1 ≤ n ‖ x ‖ ∞, {\displaystyle \|x\|_{\infty }\leq \|x\|_{1}\leq n\|x\|_{\infty },} That is, ‖ x ‖ ∞ ≤ ‖ x ‖ 2 ≤ ‖ x ‖ 1 ≤ n ‖ x ‖ 2 ≤ n ‖ x ‖ ∞. {\displaystyle \|x\|_{\infty }\leq \|x\|_{2}\leq \|x\|_{1}\leq {\sqrt {n}}\|x\|_{2}\leq n\|x\|_{\infty }.} If the vector space is a finite-dimensional real or complex one, all norms are equivalent. On the other hand, in the case of infinite-dimensional vector spaces, not all norms are equivalent.

Equivalent norms define the same notions of continuity and convergence and for many purposes do not need to be distinguished. To be more precise the uniform structure defined by equivalent norms on the vector space is uniformly isomorphic.