Once you get away from vector spaces of finite dimension, it is no longer true that every linear map is automatically continuous. For example, let Coo {f €F(N, R) f(n) = 0 for all sufficiently large n}, where F(N, R) is the vector space of all func tions from N to R with the usual pointwise operations. Another way of expressing the condition for f to belong to Coo is that {n: f(n) #0} is finite. Then define L: Coo R by L(f) = Enf(n). Notice that the sum on the right hand side is in fact a finite sum. Now (a) Check that L is linear.

Once you get away from vector spaces of finite dimension, it is no longer true that every linear map is automatically continuous. For example, let Coo {f €F(N, R) f(n) = 0 for all sufficiently large n}, where F(N, R) is the vector space of all func tions from N to R with the usual pointwise operations. Another way of expressing the condition for f to belong to Coo is that {n: f(n) #0} is finite. Then define L: Coo R by L(f) = Enf(n). Notice that the sum on the right hand side is in fact a finite sum. Now (a) Check that L is linear.

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter6: Vector Spaces

Section6.4: Linear Transformations

Problem 34EQ

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