We develop the left-definite analysis associated with the self-adjoint Jacobi operator A ( , ) k , generated from the classical secondorder Jacobi differential expressionin the Hilbert space L 2 , (−1, 1) := L 2 ((−1, 1); w , (t)), where w , (t) = (1 − t) (1 + t) , that has the Jacobi polynomials {P ( , ) m } ∞ m=0 as eigenfunctions; here, , > − 1 and k is a fixed, non-negative constant. More specifically, for each n ∈ N, we explicitly determine the unique left-definite Hilbert-Sobolev space W ( , ) n,k (−1, 1) and the corresponding unique left-definite selfadjoint operator B ( , )

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