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Aug
26
comment Conditions for Riemann integrability
You would like to know why what is true? This is the definition of Riemann integrable. Do you have another definition in mind?
Aug
22
comment Can a single point in a manifold be seen as a sub manifold?
Why would a single point not be a manifold?
Aug
14
comment Is exterior algebra an example of an algebra over a field?
@IberêKuntz, the exterior algebra is an algebra and is closed under the wedge product operation. I think you need to review what the "exterior algebra" of $V$ actually is.
Aug
14
comment Counterexample Of Banach Fixed Point (Banach's Contraction) Theorem
In order for the result to hold under your second condition you need to assume $X$ is compact and not merely complete.
Aug
14
comment Reduced Row Echelon Form with a Variable
It would have been better to put the result into the third row rather than the second as you did. Or, you can now simply switch the second and third rows. From the resulting form you should be able to determining when there will be a unique solution. You can read about "Gaussian elimination" to learn more about this type of procedure.
Aug
14
comment What, and how can, topological invariants can be computed from a space's algebra of functions?
This is relevant: mathoverflow.net/questions/82708/…. In particular, the number of connected components comes from the number of idempotent elements $C_0(X)$.
Aug
14
comment Reduced Row Echelon Form with a Variable
Keep going: do $4R_2+kR_3$
Aug
14
comment Various definitions of “topological immersion”
This is relevant: math.stackexchange.com/questions/1023163/…
Aug
14
comment Various definitions of “topological immersion”
A smooth immersion is not a local diffeomorphism: immersion here just means that the induced map on tangent spaces is injective, while "local diffeomorphism" would require that the induced map be bijective.
Aug
14
answered Sums of vector space and dimension
Aug
14
comment Sums of vector space and dimension
Yes, it only makes sense to add subspaces of the same space.
Aug
14
comment Sums of vector space and dimension
No, you cannot add vectors which do not have the same number of components. You're not using the term "dimension" correctly: it does not refer to the number of components, but rather to the size of a basis. For instance, in $\mathbb R^3$, the $x$-axis has dimension $1$ and the $yz$-plane has dimension $2$.
Aug
14
comment Non-academic mathematics development?
What is this "academy" you speak of?
Aug
14
comment Sums of vector space and dimension
Why are you trying to add something from $\mathbb R$ to something from $\mathbb R^2$? $X$ and $Y$ are subspaces of the same $\mathbb R^n$ in your question.
Aug
14
comment Sums of vector space and dimension
No, the dimensions of $X$ and $Y$ do not have to be the same. The sum is defined using the addition operation on $\mathbb R^n$, why would this require that $\dim X = \dim Y$?
Aug
13
comment Verifying a Proof for Spivak's Calculus Question (Chapter 2 Problem 9)
Whether or not this is a valid proof depends on whether induction is allowed to be used. Is it possible that, depending on how the book presents it, the question the OP is trying to prove is a first step towards establishing the principle of induction.
Aug
13
comment Verifying a Proof for Spivak's Calculus Question (Chapter 2 Problem 9)
How do you know that your $B$ is the set of all natural numbers? This is the special case of your given question when $n_0 = 1$. Is this special case something you've proved previously?
Aug
12
comment coefficients of a power series
You're forgetting the factor of $\frac1{n!}$ in the coefficients. This doesn't matter for $c_0$ and $c_1$ since $0! = 1! =1$.
Aug
12
comment Taylor Expansion for a two-variable function
Yes, this looks good.
Aug
12
comment Taylor Expansion for a two-variable function
Still not correct. The $f(x,y)$ should be $f(0,0)$, each of the partial derivatives should also be evaluated at $(0,0)$, and each of the second-order terms should be multiplied by $\frac12$.