(Solution) - In a finite dimensional normed linear space any two norms are -(2025 Original AI-Free Solution)

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In a finite-dimensional normed linear space, any two norms are equivalent.
One implication of the equivalence of norms in a normed linear space is that if a sequence converges with respect to one norm, it will converge in every norm. Therefore convergence in a finite-dimensional normed linear space is intrinsic to the sequence, and it does not depend on any particular norm. A useful corollary of this fact is given in the following exercise.