Question

State if the following statements are true or false with a brief justification

a. Any non zero bounded linear functional is an open map

b. Every bounded linear map on a complex banach space has an eigen value.

Answer 1

A non-zero bounded linear functional is not necessarily an open map. Every bounded linear map on a complex Banach space has at least one eigenvalue.

Step-by-step explanation:

a. The statement is false. A non-zero bounded linear functional is not necessarily an open map. For example, consider the linear functional defined on the real numbers by f(x) = x. This functional is bounded (since it is linear, its graph is a straight line) but it is not an open map, as it maps all open intervals to open intervals.

b. The statement is true. Every bounded linear map on a complex Banach space has at least one eigenvalue. This is known as the Spectral Theorem. It can be proven using functional analysis techniques.