Description

Linear Operators
Assignment 5
Exercise 5.1
Let (U,∥·∥U) and (V,∥·∥V ) be finite-dimensional vector spaces and L: U → V a linear map. Show that L is
bounded.
(4Marks)
Exercise 5.2
Define the multiplication operator
T : C([0,1]) → C([0,1]), (Tu)(x) = x · u(x),
Show that T is linear and bounded. Find ∥T∥ and prove your result. (4Marks)
Exercise 5.3 x ∈ [0,1].
We denote the vector space of continuously differentiable functions on the interval [a,b] by C1([1,b]). Show that

defines a norm on C1([1,b]). Is the map
T : C1([0,1]) → C([0,1]),
continuous if C([0,1]) is endowed with the ∥·∥ norm? Prove your statement or provide a counterexample.
∞
(2Marks)