Normed vector space

Equivalence of norms

Given a vector space , two norms are said to be equivalent iff there exist such that

for all . linalg This defines an Equivalence relation on norms.

Proving equivalence on the unit sphere

Since the above equation always holds for , we may divide by to get

for all with .

Properties

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

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