/sci/ - Science & Math

>>9403657
>[math]V_n^{**} = V_n[/math]
Only if [math]V_n[/math] is finite dimensional. Also replace [math]n[/math] by [math]x[/math] to denote tangent vector spaces [math]T_xM[/math] at the point [math]{\bf x}[/math], on which this [math](r,s)[/math]-tensor business makes sense.
>[math]{\bf e}_i[/math]
What space is this basis for? [math]T[/math] should act on the basis for [math]V_n[/math] and the basis for [math]V_n^*[/math] at the same time, not separately. You need to put what you've written down together.
>[math]{\bf T} \in \operatorname{End}(V_n)
\cup \operatorname{End}(V_n^*)[/math]
No. You need the exterior algebra.
>>9403776
>How is this possible?
Have you never looked at a photograph in your life before?
>>9404429
General Volterra equations of the first kind can't be cast into an ODE tho. One way to construct solutions is via the resolvent map, which exists depending on the topology of your function space.
>>9404439
Expand [math]\frac{1}{\Gamma}[/math] in its Laurent series and you'll see that at specific points [math]x \in \mathbb{R}[/math] it acquires infinitely many negative powers.
>Is there a formula for converting between this form and a polynomial series?
You mean formal Laurent series? Use the Cauchy integral formula.