How Does Finite Dimensionality Affect the Spectrum in a Banach Algebra?

[Tien]

Homework Statement

How to show element of finite dimensional banach algebra has finite spectrum?

Homework Equations

spectrum(x) = set of complex numbers 'c' with cI-x not invertible, I is identity

The Attempt at a Solution

please help to start, I don't know

 

[Mark44]

According to the rules of this forum, you have to show an effort at solving this before we can provide help.

 

[Dick]

Try showing that the eigenvalues belonging to different elements of the spectrum are linearly independent.