Frank Science
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 Nov 22 accepted Can a smooth vector-valued function approximated by functions taking value in a dense subspace? Nov 22 asked Cauchy-Lipschitz (or Picard-Lindelöf) theorem for Banach spaces? Nov 21 comment Find a factorization for $P(z)=z^5+z+1$ with $z \in \mathbb{C}$. Another way to do that is: $z^5+z+1=(z^5+z^4+z^3+z^2+z+1)-(z^4+z^3+z^2)$, where the first portion is $(z^6-1)/(z-1)=(z^3-1)(z^3+1)/(z-1)=(z^2+z+1)(z+1)(z^2-z+1)$, and the second portion is $z^2(z^2+z+1)$. Nov 21 answered Evaluating integral using invalid substitution Nov 21 comment $f \in C^2(0,\infty)$ , then how to show that $\Big(\sup_{x>0} |f'(x)|\Big)^2\le 4\big(\sup _{x>0}|f(x)|\big)\big(\sup_{x>0}|f''(x)|\big)$ ? Nov 21 revised Question about the proof of simultaneous diagonalization of quadratic forms added 1 character in body Nov 21 comment Prove that $f$ is continuous over $\cup _{i=1}^m F_i$ Hint: Fix $\epsilon>0$, for each $1\le k\le m$, you can choose $N_k$ such that for each $n>N_k$ satisfying $x_n\in F_k$, we have $d(f(x_n),f(x))<\epsilon$. Nov 21 comment $f \in C^2(0,\infty)$ , then how to show that $\Big(\sup_{x>0} |f'(x)|\Big)^2\le 4\big(\sup _{x>0}|f(x)|\big)\big(\sup_{x>0}|f''(x)|\big)$ ? When I was learning first year analysis (approximately a reduced version of Grigorii Fichtenholz's old book Differential and Integral Calculus), it gave me an impression that Taylor's formula is the only way to link derivatives. Recently I needed a similar result (I don't need the best constant), I reviewed this and several questions came into my mind without solution. Still, I don't know $L^p$-Gårding's inequality. Nov 21 comment $f \in C^2(0,\infty)$ , then how to show that $\Big(\sup_{x>0} |f'(x)|\Big)^2\le 4\big(\sup _{x>0}|f(x)|\big)\big(\sup_{x>0}|f''(x)|\big)$ ? For example, a special choice of $\phi$ leads to the Taylor formula with a remainder of integral form. There are many other choices, and I cannot see any reason why (I mean, the underlying reason) Taylor formula gives the best constant (as claimed in wikipedia). And for related interpolation inequalities, such as Caccioppoli's inequality for Laplacian or general second order elliptic operators, the choice of $\phi$ is usually something related to $f$. Nov 21 comment $f \in C^2(0,\infty)$ , then how to show that $\Big(\sup_{x>0} |f'(x)|\Big)^2\le 4\big(\sup _{x>0}|f(x)|\big)\big(\sup_{x>0}|f''(x)|\big)$ ? It was very confusing to me how Taylor formula plays a role in this, since it's usually a tool to approximate a smooth function by a polynomial locally. De facto, in order to connect the information of $f,f',f''$, we can consider $\int_a^bf''\phi$ for a test function $\phi$, then integral by part twice. Nov 21 comment Show that vector space has no countable base The key point is that, for infinite dimensional vector space $V$ over $k$, we have $\lvert V\rvert=\lvert k\rvert\cdot\dim V$. (If one cannot prove it on oneself, see mathoverflow.net/a/49572) Nov 20 comment Show that $\{f_n \} \to f$ in $L^p(E)$ iF $\{f_n\}$ belongs to and is bounded as a subset of $L^{p+\theta} (E)$. See my answer for hard-analytic viewpoint. Nov 20 answered Show that $\{f_n \} \to f$ in $L^p(E)$ iF $\{f_n\}$ belongs to and is bounded as a subset of $L^{p+\theta} (E)$. Nov 20 comment Show that $\{f_n \} \to f$ in $L^p(E)$ iF $\{f_n\}$ belongs to and is bounded as a subset of $L^{p+\theta} (E)$. Another comment: in order to prove for convergence in measure, you needn't go back to convergence a.e. In fact, do a truncation in value: for small values we use dominated convergence theorem, and for big values we use Tchebychef inequality. Nov 20 comment Show that $\{f_n \} \to f$ in $L^p(E)$ iF $\{f_n\}$ belongs to and is bounded as a subset of $L^{p+\theta} (E)$. Long ago I did this exercise, and essentially it's related to uniform integrability. Now I'm just trying another harder viewpoint, which is maybe useful in many sense, such as estimation a priori in PDE, and interpolations between Sobolev spaces. (So I don't care about easiness.) The phenomenon that bounded in an index and convergence in another index, seems pretty common to deduce some regularity of weak solutions. Nov 19 comment Show that $\{f_n \} \to f$ in $L^p(E)$ iF $\{f_n\}$ belongs to and is bounded as a subset of $L^{p+\theta} (E)$. Is it possible to introduce dyadic decomposition to solve this? (I mean, for example, decompose $x=\sum_{n\ge0}\phi_n(x)$ for $x\ge0$ where $\phi_0(x)=\min(1,x)$ and $\phi_n(x)=\min(2^n,x)-\min(2^{n-1},x)$ and extend them to odd functions). In addition, I think the original statement is true if we replace convergence a.e. by convergence by measure, or even some weaker condition. Nov 19 comment Can a smooth vector-valued function approximated by functions taking value in a dense subspace? Seems to work for general $k$. I need another time to check the details. Thanks! Nov 19 revised Can a smooth vector-valued function approximated by functions taking value in a dense subspace? added 55 characters in body Nov 19 comment Can a smooth vector-valued function approximated by functions taking value in a dense subspace? @PhoemueX Thanks very well. That's not implicit, but my mistake. Nov 19 comment Can a smooth vector-valued function approximated by functions taking value in a dense subspace? What kind of vector-valued integral are you using? I'm unfamiliar with that theory and what I can recall is only weak-integral defined in Rudin.