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I want to clear a point that "Why boundary conditions are important in taking inverse of any differential operator (lets say Laplace operator )?". What i understood is that any transformation is invertible iff its kernel is zero. If we do not specify boundary conditions then we could have any open subspace of solutions and then kernel of transformation might not be zero and hence inverse would not be possible. Could you please tell that am i thinking right or i might be missing something?

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