Equivalence of norms

Given a vector space $(𝑉, 𝕂)$ , two norms $𝑝, 𝑞 : 𝑉 \to ℝ$ are said to be equivalent iff there exist $𝑏 \geq 𝑎 > 0$ such that

𝑎 𝑞 (𝑣) \leq 𝑝 (𝑣) \leq 𝑏 𝑞 (𝑣)

for all $𝑣 \in 𝑉$ . #m/def/linalg This defines an Equivalence relation on norms.

Proving equivalence on the unit sphere

Since the above equation always holds for $𝑣 = 0$ , we may divide by $𝑞 (𝑣)$ to get

𝑎 \leq 𝑝 (𝑢) \leq 𝑏

for all $𝑢 \in 𝑉$ with $𝑞 (𝑢) = 1$ .

Properties

#state/tidy | #lang/en | #SemBr