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    Every metric space is Hausdorff

    The usual proof of this theorem seems to assume that the topology of the metric space is the one generated by the metric. But if I use another topology, for example the trivial, the space need not be Hausdorff but the metric stays the same. Am I missing something or is the statement of the...
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    Existence of basis

    Hi, I've been trying to prove that every vector space has a basis. So starting from the axioms of vector space I defined linear independence and span and then defined basis to be linear independent set that spans the space. I was trying to figure out a direct way to prove the existence of...
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    Gravitational potential energy

    Homework Statement Derive the gravitational potential energy from Newton's law. Homework Equations \mathbf{F} = -G \frac{m_1m_2}{\left|\mathbf{r}\right|^3} \mathbf{r} W = \int_A^B \mathbf{F} \cdot{} d\mathbf{r} \Delta{}U = W The Attempt at a Solution I can use the scalar version of...
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    Changing basis

    Hi, I'm reading Shankar's Principles of QM and I find it not very clear on how exactly should I change basis of operator. I know how to change basis of a vector so can I treat the columns of operator matrix as vectors and change them? Or is it something else?