I have thought about this some more and have become convinced that the following identification is correct:
Let X be a complex manifold and (TX,J) be the real tangent space with canonical complex structure. Then (TX,J) is isomorphic to the holomorphic tangent bundle (T_X, i) where the...
So I'm trying to understand the statement: On a complex manifold with a hermitian metric the Levi-Civita connection on the real tangent space and the Chern connection on the holomorphic tangent space coincide iff the metric is Kahler.
I basically understand the meaning of this statement, but...
The space of continuous functions is complete with respect to the second distance function. But what you're trying to show is that the space of continuously differentiable functions is not complete with respect to that norm. Since every continuously differentiable function is continuous, you...
For the counterexample, can you come up with a sequence of differentiable functions that converges to a non-differentiable function? Hint: Choose a very simple function that's continuous, but not differentiable.
If X_1,X_2 are simply-connected, then by definition \pi_1(X_1) = \pi_1(X_2) = 0 which certainly will cause difficulties if you're trying to prove they are isomorphic to Z.
Yet another even more geometric way of looking at it would be just straight integrate your unknown function and the "average" function. If you assume (like my last post) that f' is always less than the average or always greater than the average, then the integrals of the f and the average...
Another way of thinking about it is this. Let's say that there were some function f (that fulfills all necessary requirements for MVT to hold) such that MVT didn't hold. Then since f' is continuous (or assume that if that isn't in the hypotheses, since it's always true in "normal" phenomena), it...
Can a space be path-connected and not locally path-connected? (To be clear, "locally path-connected" just means that there is a basis of path-connected of sets.)
My general intuition says no, but my intuition seems to usually be wrong...and this would explain why Hatcher keeps referring to...
Are you asking if this converges:
\lim_{n \to \infty} \frac{(-1)^n}{n}
Look at the first few terms:
\frac{-1}{1}, \frac{1}{2}, \frac{-1}{3}, \frac{1}{4}, \ldots
Do those numbers tend to get close to something?
Other people here seem to ask if you're wondering if this converges...
Another similar question might be: Can anyone come up with an explicit Hamel basis of an infinite dimensional (separable) Hilbert space? I don't think it's possible (in most meanings of "explicit"). I mean basically the reason you throw analysis into separable infinite dimensional Hilbert spaces...
Alright that's about as complicated as I expected it to be...I basically had that written down, but apparently I don't quite have a fully functioning brain and for some reason couldn't see it. Many thanks.