Graduate Infinite matrices and the Trace function

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The discussion centers on the properties of the trace function for infinite matrices, questioning whether the identity Tr(AB) = Tr(BA) holds in this context. Participants reference the definition of the trace in linear algebra and the concept of trace class operators to explore potential answers. There is uncertainty regarding the applicability of finite matrix properties to infinite matrices. The trace class operator is highlighted as a relevant concept for understanding this issue. Overall, the conversation emphasizes the complexities of extending finite matrix properties to infinite cases.
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For finite matrices ##A## and ##B## we have
Tr(AB)=Tr(BA)
What happens in case of infinite matrices?
 
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Sorry but I am not sure that there is an answer on my question.
 
Thread 'How to define a vector field?'

Thread 'How to define a vector field?'

Hello! In one book I saw that function ##V## of 3 variables ##V_x, V_y, V_z## (vector field in 3D) can be decomposed in a Taylor series without higher-order terms (partial derivative of second power and higher) at point ##(0,0,0)## such way: I think so: higher-order terms can be neglected because partial derivative of second power and higher are equal to 0. Is this true? And how to define vector field correctly for this case? (In the book I found nothing and my attempt was wrong...
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