Aug
28
awarded Enlightened
Aug
28
awarded Nice Answer
Aug
26
comment Show that the mapping $f → f~'$ from $C^1([0 , 1])$ to $C([0 , 1])$ is not continuous.
Try functions of the form $f(x)={\sin nx\over n}$.
Aug
26
comment Example of a pointwise convergent functional sequence that is not compactly convergent.
$\chi_A$ is the function taking value $1$ on $A$ and $0$ otherwise; so, $f_n(x)=\cases{ x^n, &$x\in[0,1)$\cr 0, & otherwise }$. I usually denote sequences with parentheses $(f_n)=\{ f_n\}_{n\ge0}$.
Aug
26
comment Example of a pointwise convergent functional sequence that is not compactly convergent.
Take $f_n(x)=\chi_{[0,1)}\cdot x^n$. $(f_n)$ converges pointwise to $0$ but does not converge uniformly on the compact set $[0,1]$.
Aug
23
comment how to integrate (x-1)/(x+1)
First, do the division.
Aug
23
comment Why is convexity of $S$ needed in these questions?? (Bachman &Narici, Q-17,18)
For your second concern, see this.
Aug
23
comment Why is convexity of $S$ needed in these questions?? (Bachman &Narici, Q-17,18)
Consider $( (-1)^n)$ in $\Bbb R$ with $S=\{1,-1\}$.
Aug
22
revised Let $f:M\to N$ be continuous, then $f(U)\subset V$.
added 1 character in body; edited tags
Aug
18
comment IF $\lim_{n\to\infty}a_{n}=l$, Then prove that $\lim_{n\to\infty}\frac{a_{1}+a_2+\cdot..+a_n}{n}=l$
See this and its "linked' section.
Aug
17
comment How to solve $\lim_{n\to \infty}\sin(1)\times \sin(2)\times\sin(3)\times\ldots\times\sin(n)$
Hmm. From my first comment, at least $1/4$ of the terms are bounded by $\sqrt2/2$ in absolute value. You can use this to show the limit is $0$ (but the argument referenced in my second comment is nicer).
Aug
17
comment How to solve $\lim_{n\to \infty}\sin(1)\times \sin(2)\times\sin(3)\times\ldots\times\sin(n)$
For 2, you could use the first answer here (and may as well use it for 1.).
Aug
17
comment How to solve $\lim_{n\to \infty}\sin(1)\times \sin(2)\times\sin(3)\times\ldots\times\sin(n)$
For 1, note each interval $[n\pi-{\pi\over4}, n\pi+{\pi\over4}]$, contains an integer.
Aug
16
comment Subsequences and blocks of Schauder bases
You can peturb the standard unit vector basis $(e_n)$ of $\ell_1$ to obtain a basis $(f_n)$ of $\ell_1$ in such a way that each $f_j$ has full support. If $Y$ contains an $f_j$ and if a subsequence of $(e_n)$ is a basis of $Y$, then $Y$ would have to contain every $e_j$.
Aug
15
comment locally uniform convergence vs pointwise convergence
You might find this interesting.
Aug
15
comment How can the harmonic series diverge if at large n you keep adding terms close to zero?
It may be easier to consider $1+{1\over2}+{1\over2}+{1\over3}+{1\over3}+{1\over3}+\cdots$. The terms converge to $0$, but the sum diverges, since "chunks" of it sum to $1$.
Aug
14
comment Prove the following inequality envolving $L^{1}$ and $L^{2}$ norms
Holder.${}{}{}{}$
Aug
12
comment Line segment in the unit sphere
You need to find points $x$ and $y$ in $S_X$ such that $\lambda x+(1-\lambda) y$ is in $S_X$ for all $0\le \lambda\le 1$.
Aug
12
comment Can We Always Realize the Value of the Quotient Norm.
You're welcome.
Aug
12
comment Can We Always Realize the Value of the Quotient Norm.
No. See this.
1 2 3 4 5