The Hilbert Basis Theorem

Proof. The Hilbert basis theorem (A.5) shows that any nonempty collection... 

Proof. The Hilbert basis theorem (A.5) shows that any nonempty collection of ideals in k[X, X ] has a maximal element hence any nonempty... 

If the Hellmann-Feynman theorem is to be valid for forces on nuclei, the Coulomb cusp condition must be satisfied. However, if the nuclei are displaced, the orbital Hilbert space is modified. Hurley [179] noted this condition for finite basis sets, and introduced the idea of floating basis functions, with cusps that can shift away from the nuclei, in order to validate the theorem for such forces. 

If the Hilbert space of the variational basis set is invariant under scale transformation, then the hypervirial theorem implies... 

The geometrical structure of the Hilbert space makes it possible to build a basis in the Hilbert space, similar to the orthogonal basis in the Euclidean space. We will introduce a basis by the following sequence of definitions and theorems. 

Our first way of answering the last question will be based on the fundamental theorems on Hilbert space [14], Indeed, the theorem on separability tells us that any subspace of h is also a separable Hilbert space. As a consequence, the inner product defined on, say, the occupied subspace is hermitian irrespectively of the choice of the basis x f (/)], as long as this latter satisfies the fundamental requirements of Quantum Mechanics. One should therefore not have to impose this property as a constraint when counting the number of conditions arising from the constraint CC+ =1 but, on the contrary, can take it for granted. 

The connection between the two Hilbert spaces and L2 used in physics is established by using the existence of at least one orthonormal basis (p = (pk(X) in L2, which leads to the expansion theorem... 

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