### Questions (103)

 9 Show that Hill's Equation $u'' + a(t)u=0$ if $a(t)<0$ for all $t$ then $u\to\infty$ as $t\to\infty$ 6 Let $\lim\limits_{n\to\infty}a_{n+1}-a_{n}=\alpha$. Show that $\lim\limits_{n\to\infty} \frac{a_n}{n}=\alpha$. [duplicate] 5 $X$ and $Y$ Banach Spaces, $T \in B(X,Y)$, $Y = \operatorname{im}T \oplus M$, for $M \subseteq Y$, then $\operatorname{im}T$ is closed in $Y$ 5 Schur's Theorem: In $\ell^1$ weak convergence of $x_n$ is the same as convergence in the norm 4 No non-negative continuous function on $[a,b]$ such that $\int_a^b f(t)dt=1, \int_a^b tf(t)dt=c, \int_a^b t^2f(t)dt=c^2$ for $c\in\mathbb{R}$.

### Reputation (1,159)

 +11 Bounded linear operator $A$ s.t. $Ax=y$ has a least square solution for each $y$ iff the range of $A$ is closed +30 1 and 2 norm inequality +5 Show that $T$ is self adjoint where $Tf(x) := \int k(x,y)f(y)\,dy$ when $k(y,x) = \overline{k(x,y)}$. +5 Finding the flow of vector field with lie group (orthogonal group)

### Answers (38)

 7 Prove $f(x) =\ \frac{x^5}{5!} +\frac{x^4}{4!} +\frac{x^3}{3!}+\frac{x^2}{2!} +x+1$ has only one root. 6 1 and 2 norm inequality 5 How form a linear mapping $f:\mathbb{R}^{4} \rightarrow \mathbb{R}^{3}$ to a matrix correctly?(solved) 4 How to add powers? 3 Equality/Equivalence of functions

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 11 derivatives × 3 6 analysis × 15 9 linear-algebra × 17 6 numerical-methods × 4 9 integration × 7 6 norm × 2 8 matrices × 4 6 linear-transformations × 2 8 vectors × 3 5 abstract-algebra × 3

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