Normed vector space

Equivalence of norms

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

𝑎𝑞(𝑣)≤𝑝(𝑣)≤𝑏𝑞(𝑣)

for all 𝑣 ∈𝑉. 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≤𝑝(𝑢)≤𝑏

for all 𝑢 ∈𝑉 with 𝑞(𝑢) =1.

Properties

• All norms on a finite dimensional space are equivalent
• Norms are equivalent iff they induce the same topology.

---

tidy | en | SemBr