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2d
revised Can you equip every vector space with a Hilbert space structure?
added more material in response to a comment
2d
comment Can you equip every vector space with a Hilbert space structure?
@Sushil I'll consider the converse a little later, when I have more time and leisure. For the other question, if $\binom{B}{\alpha}$ denotes the collection of subsets of $B$ of size $\alpha$, then by taking graphs of functions we have an injection $A^\alpha \to \binom{\alpha \times A}{\alpha}$, and since $|\alpha \times A| = |A|$ when $\alpha \leq |A|$, there is a bijection $\alpha \times A \to A$, and by composition along this bijection we get an injection $A^\alpha \to \binom{A}{\alpha}$.
Jun
27
awarded  Enlightened
Jun
27
awarded  Nice Answer
Jun
26
comment What is the significance of having Prime Ideal Theorem in models for failure of Axiom of Choice?
I'm not sure the question would be such a duplicate. OP does not say she is looking for a bunch of consequences, but for the scope of set-theoretic tools made available under this weaker form of choice. One such is the ultrafilter lemma (one can choose an ultrafilter extending any proper filter), which is quite a powerful tool. Such a tool is used for example to construct ultraproducts, in the compactness theorem, and so on. I think a more creative answer might be possible, here at MO.
Jun
20
comment a Problem about Harmonic Function
Crossposted to MO: mathoverflow.net/questions/209744/…
May
31
comment If there is in a category $\mathcal{A}$ finite products and equalizers then it has pullbacks
Moderators can export comments, and I can do so for you, but you should open an account there. But a hint for your problem is to see the pullback of two maps $f: A \to C$ and $g: B \to C$ as a subobject $e = \langle p_1, p_2 \rangle: P \to A \times B$ where $e$ is an equalizer of two maps $A \times B \to C$ where the relevant equation expresses the commutativity of the pullback diagram. But this is a very good exercise for you to do on your own.
May
29
comment If there is in a category $\mathcal{A}$ finite products and equalizers then it has pullbacks
Alice, please do not ask these homework questions here. It was already explained in a comment on your other question that we do not handle such questions here; this site is for research of professional mathematicians. Your question would be more appropriate for mathematics.stackexchange.
May
11
awarded  Necromancer
May
4
revised Can you equip every vector space with a Hilbert space structure?
small correction; added a proof; incorporated a comment and added a link
Apr
11
comment What's the latest on Laver tables?
I'm wondering whether it's provable in ZFC that there is a row with period 32 even. The tables are astonishing indeed.
Mar
27
comment Proving that $(e^x+1)^{1/3}$ has no elementary antiderivative
meta.mathoverflow.net/questions/1700/…
Mar
3
revised Integral $\int\frac{dx}{x^5+1}$
fixed a minor mathematical error
Mar
3
answered Integral $\int\frac{dx}{x^5+1}$
Feb
10
comment discontinuity of a monotonic function
(1) I had to zoom in at least six times before I could even read the quote. (2) Deal with what problem? I don't see a problem.
Feb
10
answered Dimension of Hom(U,V)
Feb
8
comment How to write $\sin^2(z)$ in terms of imaginary units.
You probably want an $x$ instead of a $z$.
Feb
5
answered $A$ and $B$ are finite sets. How many Partial Functions exist between them?
Feb
3
answered What's the difference between an axiomatization and a characterization of a structure?
Feb
2
comment Ring functions which 'respects' homomorphisms
It means that I don't visit this site regularly (and so might not notice your post when it appears). "Tune in" is mainly a radio or TV reference, as when you go to a specific channel at a specific time to "tune into" a show. "M.SE" is short for Mathematics StackExchange.