1h
comment Check if set of functions is a basis of space
Yes. But it's easy to do. Note any finite linear combination of your vectors gives a function that takes on only finitely many distinct values. (So, it's not a basis.)
1h
comment Check if set of functions is a basis of space
Consider a function that takes on infinitely many distinct values.
1h
comment How to check the compactness of these sets:
Each set contains the standard unit vectors. Does this set have a limit point?
1h
comment Check if set of functions is a basis of space
Note you need to check if any given function is a finite linear combination of the $f_\alpha$...
1h
comment How to prove that $\lim_{n\to\infty}{|\sin n|}$ doesn't exist
Hint: Each interval $[n\pi/2-1/2,n\pi/2+1/2]$ contains an integer.
1d
comment Independence in Banach space
A trivial example: take $\ell_1(\Bbb R)$ and the standard unit vectors $\{e_\alpha:\alpha\in\Bbb R\}$. I'm not sure what happens in the separable case; but I'll think about it.
1d
comment suppose that f is a positive continuous function on R^n such that lim as abs(x) goes to infinity f(x)=0. Show that f attains its maximum
First choose a closed ball, $B$, about $0$ so that $\Vert f\Vert$ is at most $\Vert f(0)\Vert$ on $B^C$.
1d
comment How can I prove that a sequence such that every converging subsequence coverges to the same limit, converges?
For example the sequence $(x_n)$ with $x_n=n$ satisfies your hypothesis with $a=123833$.
1d
comment How can I prove that a sequence such that every converging subsequence coverges to the same limit, converges?
Not if the sequence is unbounded.
Nov
19
comment Let $f$ a continuous function on $\mathbb R$ such that $f(0)=f(2)$.
See this.
Nov
19
comment Is $A$ a compact operator or not?
@OEmpaladordeCabras Yes, thanks for pointing out the typo.
Nov
19
comment Scaling axis on Mathematica's plots
Mathematica has its own site.
Nov
19
answered Continuous vs. Monotone Functions
Nov
19
comment Continuous vs. Monotone Functions
No. The Weierstrass nowhere differentiable function is monotonic on no interval, e.g..
Nov
19
comment Prove that there is no Hilbert basis in an innite dimensional Hilbert space which is a linear basis.
If $(e_n)$ were a normalized Hilbert basis, then $x=\sum e_n/2^n$ would be an element of the space for which $(x,e_n)\ne0$ for all $n$.
Nov
19
comment Show that f is uniformly continuous on [0, +∞)
Use the fact that $f$ is uniformly continuous on $[0,a+1]$.
Nov
17
comment Why is $x$ restricted this way? (limits of functions)
Oh, I missed that.
Nov
17
comment Why is $x$ restricted this way? (limits of functions)
"$x\rightarrow a^+$" means $x$ is approaching $a$ from the right (so, with $x>a$).
Nov
17
comment Continuous and preserves measurability $\implies$ preserves null sets.
Hint: A set of positive measure contains a non-measurable subset, and any subset of a null-set is a null-set.
Nov
17
comment Convergence of a Sum corresponding to the Banach Space $c_0$
See this.
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