1d
comment A question on infinite series and boundedness of sequence
If $(a_n)$ is unbounded, you can choose a subsequence $(a_{n_k})_k$ with $|a_{n_k}|>k^2$ for each $k$. Consider the sequence $(x_m)$ with $x_{m}=1/k^2$ if $m=n_k$ and $x_m=1/ 2^m $ otherwise.
1d
comment Show a open set has no minimum
Hint: If $A$ is open and $x\in A$, there is a $\delta>0$ such that $(x-\delta,x+\delta)\subseteq A$. In particular, then, $x-\delta/2\in A$.
1d
comment Trigonometry Identity: $\tan \theta\sin \theta + \cos \theta = \sec \theta$
First write $\tan\theta\sin\theta+\cos\theta={\sin^2\theta\over\cos\theta}+{\cos^2\theta \over\cos\theta}$.
2d
comment If $T: X \to Y$ is norm-norm continuous then it is weak-weak continuous
My previous comment addressed the question in the post body. I assumed (probably incorrectly) that you were referring to norm-continuity. For the question in your title, you could show that if a net $(x_\alpha)$ converges weakly to $x$, then $(Tx_\alpha)$ converges weakly to $Tx$. Towards this end, note $y^*\circ T\in X^*$ if $y^*\in Y^*$ (I presume you already did this for sequences; the proof for nets is the same).
2d
comment If $T: X \to Y$ is norm-norm continuous then it is weak-weak continuous
Yes. See this.
Aug
15
comment Measure Theory Inequality
Lebesgue measure is a regular measure.
Aug
15
comment Measure Theory Inequality
You can find an open set $O$ containing $I\setminus A$ with $\mu (A\cap O)<\mu(I\setminus A)\epsilon $. Note $O$ can be written as a disjoint union of countably many non-degenerate open intervals. Show that your inequality holds for at least one of these intervals.
Aug
14
comment Seperating neighborhoods of infinite sets in normal topological spaces
See the answer to this post.
Aug
14
comment Calculus - esplion-delta prove
Hint: $\cos \theta>1/2$ for $0\le\theta\le \pi/3$.
Aug
14
comment GRE - Probability Question
If $B\subset A$, then $P(A\cup B)=P(A)=1/2$ (you could also just observe that the probability of the union is greater than both $P(A)$ and $P(B)$). If $A\cap B=\emptyset$, then the probability of the union is the sum $P(A)+P(B)$.
Aug
14
comment Help understanding Archimedes method for finding the area of a circle
Draw the other diagonal of the square and apply the Pythagorean Theorem to the so-formed right triangles.
Aug
14
comment GRE - Probability Question
The smallest it could be is when $B\subset A$. The largest it could be is when $A\cap B=\emptyset$. Can you find the probability for these extreme cases?
Aug
14
comment Infinite set of natural numbers and the set difference
Take the set of even integers.
Aug
14
comment Prove that every subset of $\mathbb{R}$ is compact in the finite complement topology.
Looks fine. ${}$
Aug
13
revised On a Banach space $X$, is the functional $x \mapsto \frac{1}{p}\|x\|^p$ convex?
added 14 characters in body
Aug
13
revised How to prove that a diffrensiation of a formula equals to another formula.
edited tags
Aug
13
comment Radon-Riesz & Kadec-Klee
I'm not sure, but I think you may be able to modify the argument given here.
Aug
12
comment Solve: $\lim\limits_{x\to \infty} \frac{\sqrt{x+2}}{\sqrt{2x-2}}$
You ignored the square roots.
Aug
12
comment How to find a nonlinear function $f:\mathbb{R}^2\to\mathbb{R}^2$ that is almost linear in the sense $f(\alpha (a,b))=\alpha f(a,b)$?
@alex.jordan See this.
Aug
12
comment Lagrange's form of the remainder vs Cauchy's form
Here's one.
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