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seen Apr 17 at 19:49

Feb
9
accepted Extending an independent set of vectors to a basis
Feb
6
comment Extending an independent set of vectors to a basis
I think I don't use Zorn's lemma but instead I use the axiom of choice (to construct the chain $T$) so I guess both proofs rely on the same principle (since you can get Zorn's lemma assuming the axiom of choice).
Feb
6
asked Extending an independent set of vectors to a basis
Jan
4
answered Is the finite dimension required in this proof?
Jan
4
accepted Dimension of a vector space when sum and multiplication changes
Jan
4
asked Dimension of a vector space when sum and multiplication changes
Dec
6
awarded  Nice Question
Nov
17
accepted tensor product and wedge product for direct sum decomposition
Nov
9
accepted Pfaffian system of equations
Oct
8
awarded  Tumbleweed
Oct
1
asked group of automorphism of an $SU(2)$-bundle
Sep
16
comment Are injectivity and surjectivity dual?
see also: math.stackexchange.com/questions/480837/…
Sep
16
comment Are injectivity and surjectivity dual?
Assume $w'$ is linearly independent with $w$, is neither in the image of $f$, but $w'+w$ is in the image. Then $l(w'+w)$ is not $0$ so $f^\vee(l)\neq 0$. I came with this proof: $Ann \left( Im (f) \right)= \{\varphi \in W^\vee : \varphi(w)=0 \; \forall w \in Imf \}= \{\varphi \in W^\vee : f^\vee(\varphi)=0 \} = Ker f^\vee$ So $f^\vee$ injective iff $Ker f^\vee=0$ iff $Ann \left( Im (f) \right)= 0 $ iff $Im (f)=W$ iff $f$ surjective. I'm not sure if this proof is right since as stated in Tu's book "An introduction to Manifolds" we need finite dimensions for $f$ surjective
Sep
1
comment Is the finite dimension required in this proof?
@Pink Elephants you are right!
Sep
1
comment Is the finite dimension required in this proof?
It is problem 10.5 of Tu's book "An introduction to Manifolds". Yes the axiom of choice is assumed so every vector space has a basis. @Pink Elephants the only elements which are not mapped to $0$ by $\alpha_i \circ L$ are those whose image by $L$ is a multiple of $e_i$.
Aug
31
asked Is the finite dimension required in this proof?
Aug
15
awarded  Yearling
Jul
3
accepted Limit point of an infinite set in a compact space
Jul
3
asked Limit point of an infinite set in a compact space
Jun
2
accepted Application of the implicit function theorem