Real Analysis

If $f\in R(\alpha )$ and $g\in R(\alpha )$ on [a, b], then prove that $f+g\in R(\alpha )$. If $f\in R$ on [a,b] and if there is a differentiate function F on [a, b] such that $F^{\prime }=f$, then

Explain - Rearrangements of terms of a series with an example. Define a curve $\gamma$ in $R^k$. Write a short note on this curve and define its length.

Define the following

1. Uniform convergence

Define the following

1. Point - wise convergence

Define the following (Any two)

1. Linear combination
2. Independent set
3. Standard basis

Define the following :- (Any two)

1. Linear Transformation
2. Norm of A
3. Continuously differentiable function

Define the following :-

1. Power series
2. Derivatives of higher order.

Suppose ${\displaystyle \sum _{n=0}^{\infty }{C}_n}$ converges. Put $f(x)={\displaystyle \sum _{n=0}^{\infty }{C}_n{x}^n(-1<x<1)}$ then prove $\underset{x\to 1}{\lim }f(x)={\displaystyle \sum _{n=0}^{\infty }{C}_n}$.

If P* is a refinement of P, then prove: $L(P,f,\alpha )\le L(P*,f,\alpha )$. If $f_1\in R(\alpha )$ and $f_2\in R(\alpha )$ on [a, b] then prove

1. $f_1+f_2\in R(\alpha )$
2. $Cf\in R(\alpha )$ for every constant $C$.

also prove that

If $f$ maps [a,b] into $R^k$ and if $f\in R(\alpha )$ for some monotonically increasing function $\alpha$ on [a,b] then $\left|f\right|\in R(\alpha )$, and $\left|\int fd\alpha \right|\le \int \left|f\right|d\alpha$. If $\gamma$ is continuous on [a,b], then $\gamma$ is rectifiable, and $\wedge (\gamma )={\int }_a^b\left|{\gamma }^{\prime }(t)\right|dt$.

Suppose $<f_n>$ is a sequence of functions, differentiable on [a,b] and such that $f_n(x_0)\to f(x_{})$ converges for some point $x_0$ on [a,b]. (b) if $f_n^{\prime }$ converges uniformly on [a,b], then $<f_n>$ converges uniformly on [a,b] to a function $f$, and $f^{\prime }(x)=\underset{n\to \infty }{\lim }{f}_n^{\prime }(x)$. (a) If $S_n=\frac{m}{m+n}$ for $m=1,2,3\dots$ and $n=1,2,3\dots$ then prove that $\underset{m\to \infty }{\lim }\underset{n\to \infty }{\lim }{S}_n=\underset{n\to \infty }{\lim }\underset{m\to \infty }{\lim }{S}_n$.

State the difference between uniform convergence and point-wise convergence.

State Cauchy criterion for uniform convergence.

Suppose E is an open set in $R^n$, $\overrightarrow{f}$ maps E into $R^m$, $\overrightarrow{f}$ differentiable at ${\overrightarrow{x}}_0\in E$, $\overrightarrow{g}$ maps an open set containing $\overrightarrow{f}(E)$ into $R^k$ and $\overrightarrow{g}$ is differentiable at $\overrightarrow{f}({\overrightarrow{x}}_0)$ then the mapping $\overrightarrow{F}$ of E into $R^k$ defined by $\overrightarrow{F}(\overrightarrow{x})=\overrightarrow{g}(\overrightarrow{f}(\overrightarrow{x}))$ is differentiable at ${\overrightarrow{x}}_0$ and ${\overrightarrow{F}}^{\prime }({\overrightarrow{x}}_0)=g^{\prime }(\overrightarrow{f}({\overrightarrow{x}}_0))f^{\prime }({\overrightarrow{x}}_0)$.

Define partial derivatives

Suppose $\overrightarrow{f}$ maps a convex open set $E\subset R^n$ into $R^n$, $\overrightarrow{f}$ is differentiable in E, and there is a real number M such that $\left|f^{\prime }(x)\right|\le M$ for every $x\in E$. for all $\overrightarrow{a},\overrightarrow{b}\in E$.

Suppose the series ${\displaystyle \sum C_n{x}^n}$ converges for $\left|x\right|<R$ and define $f(x)={\displaystyle \sum C_n{x}^n}$ for $\left|x\right|<R$. Prove that the series ${\displaystyle \sum C_n{x}^n}$ converges uniformly on $[-R+\epsilon ,R-\epsilon ]$ no matter which $\epsilon >0$ is chosen. Suppose $f(x)={\displaystyle \sum _{n=0}^{\infty }{C}_n{(x-a)}^n}$ this series converges in $\left|x-a\right|<R$. If $R>0$ then $f$ can be expanded in a power series about the point $a$ which converges in $\left|x-a\right|<R$, and $f^{\prime }(x)={\displaystyle \sum _{n=<!--</mo-->\; <mn>1</mn>}^{\infty }n{C}_n{(x-a)}^{n-1}}$.