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A Hilbert space $\cal H$ is complete which means that every Cauchy sequence of vectors admits a limit in the space itself. Under this hypothesis there exist Hilbert bases also known as complete orthonormal systems of vectors in $\cal H$. A set of vectors $\{\psi_i\}_{i\in I}\subset \cal H$ is called an orthonormal system if $\langle \psi_i |\psi_j \rangle ... 6 Like Wikipedia says: "Moment is a combination of a physical quantity and a distance." This 'physical quantity' could be various things. To take the examples you mention: Moment of momentum (commonly known as angular momentum) is expressed as$\vec{L}=\vec{r}\times m\vec{v}$, and is a measure for the rotational momentum of an object around some axis. Moment ... 3 This completeness relation of the basis means that you can reach all possible directions in the Hilbert space. It means that any$|\psi \rangle$can be made up from these basis vectors. If the sum of the projectors (the ket-bras) would not be the unit matrix, the vector$|\psi\rangle$could have components which cannot be represented within your basis. ... 2 Diffeomorphism Invariance Let$M$be a smooth manifold. Let$\phi: M \to M$be a diffeomorphism. A simple property of the Einstein equations is $$g \in \otimes^2 TM \text{ is solution to vacuum Einstein equation} \implies \text{ so is } \phi^*g$$ To see that this is true, simply pull back both sides of the Einstein equation by$\phi\$, and use the ...

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