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I've just visitted another Maths forum and picked up two interesting questions:

Question 1. Let H and K be homomorphic normed linear spaces. Is it necessary that H and K have the same dimension if both H and K are finite-dimension ? Is there possible a homomorphism between an infinite-dimension normed space and a finite-dimension one ?

Question 2. If f is a homomorphism between two normed linear spaces, is f necessary uniformly continuous ?

Hope to get your ideas.