Undergrad Hermitian Operators and Non-Orthogonal Bases: Exploring Infinite Spaces

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The discussion centers on the properties of Hermitian operators when applied to infinite-dimensional spaces, specifically using the basis {1, x, x², x³, ...}. It highlights that traditional rules for Hermitian operators, such as eigenvectors spanning the space, may not hold in infinite dimensions. The distinction is made that while the matrix of a self-adjoint operator is equal to its conjugate transpose in an orthonormal basis, this does not apply to non-orthonormal bases. The conversation emphasizes the complexities introduced by infinite spaces and the implications for Hermitian operators. Understanding these differences is crucial for accurately analyzing operator behavior in such contexts.
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It is not necessary.
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The basis he is talking about: {1,x,x²,x³,...}
I don't know how to answer this question, the only difference i can see between this hermitians and the others we normally see, it is that X is acting on an infinite space, and, since one of the rules involving Hermitian fell into decline in the infinity space (e.g Eigenvectors span the space),i am not sure if another rules changes in the infinity space too..
 
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The matrix of a self-adjoint operator with respect to an orthonormal basis is equal to its conjugate transpose. Your basis is not orthonormal.
 
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Infrared said:
The matrix of a self-adjoint operator with respect to an orthonormal basis is equal to its conjugate transpose. Your basis is not orthonormal.
Forgot this detail, thank you.
 
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