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...
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...
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...
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?