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Every operator in finite-dimensional space can be represented by a matrix in a given basis in this space.

If the transpose conjugate of the matrix representation of an operator in a given basis is the same of the original matrix representation of this operator in this basis, then this operator is hermitian.

But what confuses me is that if this applies to a given basis what guarantees that it will apply to any basis in the space?

Even in infinite dimensional spaces,for example the derivative operator D in the position basis is : <x|D|x'>=δ'(x-x'), where δ'(x-x') is the derivative of the dirac delta function, this will yield that the operator D is not hermitian. But what if the operator D is represented in another basis,will it also be non-hermitian in the other basis?

How can the operator be hermitian in general ?

I hope my question is clear.