Find the cardinality of the set of all norms on R^n (hint: show that every norm || || : R n → R is continuous).
Answer
- The questioner was satisfied with and accepted the answer, or
- The answer was evaluated as being 100% correct by the judge.
1 Attachment
![Erdos](https://matchmaticians.com/storage/user/100028/thumb/matchmaticians-3empnt-file-5-avatar-512.jpg)
-
Leave a comment if you need any clarifications.
-
Do you also need the proofs for your other question or just the answer? I would say your offered amount is a bit low.
-
Proofs, how much more do you think i should add?
-
Well, it indeed needs three proofs (one for each case ) and will take about an hour to write. I think the offer should be at least tripled.
-
Okay i'll do that
-
I suggest you to also set a later deadline, if possible. 4 hours is too early for this kind of high level questions.
-
Hey Phillip, i just realized, don't i need a lower bound for the Cardinality for the set of all norms
-
Cool. So you do not need the second part of the argument.
-
im confused. I still need a lower bound for the cardinality for the set of all norms
-
Suppose you have a norm || . || . Then for any real number a, a*|| . || is also a norm. So The cardinality of the set of norms is at least that of the continuum. I hope this makes it clear.
-
See also the attached file.
- answered
- 501 views
- $7.00
Related Questions
- Is it true almost all Lebesgue measurable functions are non-integrable?
- Prove that a closed subset of a compact set is compact.
- Define$ F : C[0, 1] → C[0, 1] by F(f) = f^2$. For each $p, q ∈ \{1, 2, ∞\}$, determine whether $F : (C[0, 1], d_p) → (C[0, 1], d_q)$ is continuous
- Show that there is either an increasing sequence or a decreasing sequence of points $x_n$ in A with $lim_{n\rightarrow \infty} x_n=a$.
- True-False real analysis questions
- A problem on almost singular measures in real analysis
- Prove the uniqueness of a sequence using a norm inequality.
- real analysis