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 Jun 4 comment About inverse matrix in portfolio choice You are right. I had a typo. In the textbook, there is no 1/2 in Equation (*). Thanks. Apr 30 comment uniform convergence of continuous functions I finished all the details but one thing: how can I prove uniform convergence. It seems that even on the compact set, pointwise convergence does not imply uniform convergence. By the way, I would accept the comments as the answer (should you post it or them), since they are very helpful. Thank you. Apr 30 comment uniform convergence of continuous functions One quick question: is it right to argue that Convex combination of continuous functions is continuous $\Rightarrow C[0,1]$ is convex. Apr 30 comment uniform convergence of continuous functions Riesz representation theorem seems to apply to linear functionals. But C[0,1] meas all continuous functions on [0,1]. Am I wrong? Apr 22 comment The spectrum of a bounded linear operator Thank you. I basically proved that. Apr 22 comment The spectrum of a bounded linear operator I agree with your EDIT''. In fact, $\sigma(T^{n})=[\sigma(T)]^{n}$ for the complex case. That's why the converse is true for the complex. The proof is not trivial. Do you agree? Also @Norbert Apr 22 comment The spectrum of a bounded linear operator One simple linear algebra question. Given that $$T^n-\lambda^nI=\left(\sum\limits_{k=0}^{n-1}\lambda^{n-1-k}T^k\right)(T-\lambda I)$$, why $T^n-\lambda^n I$ is invertible $\Rightarrow T-\lambda I$ is invertible? Thank you. Apr 19 comment The spectrum of a bounded linear operator Thank you, but I cannot see how the rotation of matrix is related to this question. Can you clarify a little? Apr 15 comment About asymmetric simple random walk I agree with your argument, but that is not the question here. I double checked the question. It is exactly the same as the textbook. But frankly speaking I find this exercise annoying, because later we only need the conclusion that it is bounded. So the upper bound such as yours is sufficient for the later use. Anyway, I can not argue that the exercise is wrong. Thank you! Apr 10 comment About the notation of the probability measures I checked the textbook. They are the same. Apr 10 comment About the notation of the probability measures Thank you. I saw this argument somewhere. So you think we have $\mu_{n}\overset{w}{\rightarrow}\mu\Leftrightarrow P^{X_{n}}\overset{w}{\rightarrow}P^{X}$ ? Apr 10 comment About the notation of the probability measures So do we have "$\mu_{n}\overset{w}{\rightarrow}\mu\Leftrightarrow P^{X_{n}}\overset{w}{\rightarrow}P^{X}$"? Apr 9 comment Construction of a special continuous function Yes. It does help. Two answers are basically the same and are provided basically at the same time. The first answer is 1 min earlier. Thank you. I mark this answer as helpful. Apr 9 comment Construction of a special continuous function Just one simple question: here the norm means the Euclidean norm. Right? en.wikipedia.org/wiki/Euclidean_norm#Euclidean_norm Apr 9 comment Construction of a special continuous function Thank you! I copied the original text. I guess it should mean for all $\epsilon>0$. I don't know how to modify the bump function to make it continuous. Apr 9 comment Weak convergence of Dirac measures Thank you, Byron. I edited the question to make it more specific. Could you please further clarify it? Thanks. Mar 16 comment Matrix norm $T:l_{\infty}^{2}\rightarrow l_{\infty}^{2}$ Thank you! It is very helpful. Mar 15 comment Hahn-Banach. Extend the functional by continuity Finally, problem solved. Thank you so much! Mar 15 comment Hahn-Banach. Extend the functional by continuity @t.b. I figured out everything except one question: I need to use $x_n\rightarrow x\Rightarrow \lim ||x_n||=||x||$. I guess it should be OK, but I feel I have not seen such a property before. Mar 15 comment Hahn-Banach. Extend the functional by continuity @t.b Thanks a lot. I guess that general fact is base on Tietze extension theorem, which I can not use without proof. So I will probably still try azarel's idea.