Proving Convexity of Numerical Range for Normal T

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The discussion revolves around proving the convexity of the numerical range of a normal linear transformation T in a finite-dimensional inner product space over complex numbers. Participants are seeking clarification on the definitions and properties involved, particularly what is meant by "normal" and the specific nature of the inner product space and the codomain of T. The main goal is to demonstrate that if two points a and b are in the numerical range of T, then any convex combination of these points, represented as (1-c)a + cb for 0 <= c <= 1, is also within the numerical range. The conversation highlights the need for a deeper understanding of the terms and concepts before proceeding with the proof. Overall, the focus is on establishing the mathematical framework necessary for the proof of convexity.
julydecember
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Homework Statement


If T is a linear transformation on the finite-dimensional inner product space over complex numbers and is normal, then prove that the numerical range of T is convex.


Homework Equations





The Attempt at a Solution


If we assume a and b are in the numerical range of T, then we have to prove that (1-c)a + c b is also in the numerical range of T for 0<= c <= 1. Can someone give some help, please?
 
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julydecember said:

Homework Statement


If T is a linear transformation on the finite-dimensional inner product space over complex numbers
which finite-dimensional inner product space? And what is the codomain of T?

julydecember said:
and is normal,
what does normal mean?

julydecember said:
then prove that the numerical range of T is convex.
Do you mean "prove that the image of T is convex"?
 

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