UMD Analysis Qualifying Exam/Jan08 Real

${\displaystyle {\begin{aligned}\lim _{M\rightarrow \infty }\int _{[-M,M]}|f|&=\int _{\mathbb {R} }|f|\\\lim _{M\rightarrow \infty }\int _{[-M,M]}|f|-\underbrace {\int _{\mathbb {R} }|f|} _{<\infty {\mbox{ since }}f\in L^{1}}&=0\\\int _{\mathbb {R} }|f|-\lim _{M\rightarrow \infty }\int _{[-M,M]}|f|&=0\\\lim _{M\rightarrow \infty }\int _{[-M,M]^{c}}|f|&=0\end{aligned}}\!\,}$

Suppose ${\displaystyle \lim _{|x|\rightarrow \infty }f(x)\neq 0\!\,}$. Then,

${\displaystyle \lim _{x\rightarrow \infty }f(x)\neq 0\!\,}$

or

${\displaystyle \lim _{x\rightarrow -\infty }f(x)\neq 0\!\,}$

Without loss of generality, we can assume the first one, i.e., ${\displaystyle \lim _{x\rightarrow \infty }f(x)\neq 0\!\,}$ (see remark below to see why this)

Note that ${\displaystyle \lim _{x\rightarrow \infty }f(x)=0\!\,}$ can be written as

${\displaystyle \forall \epsilon >0\left[\exists M>0\left[\forall x>M\left[|f(x)|<\epsilon \right]\right]\right]\!\,}$

Then, the negation of the above statement gives

${\displaystyle \exists \epsilon _{0}>0[\forall M>0[\exists x_{0}>M[|f(x_{0})|>\epsilon _{0}]]]\!\,}$

Because of the uniform continuity, for the ${\displaystyle \epsilon _{0}\!\,}$ there is a ${\displaystyle \delta (\epsilon _{0})>0\!\,}$ such that

${\displaystyle |f(x_{0})-f(x)|<{\frac {\epsilon _{0}}{2}}\!\,}$ ,

whenever ${\displaystyle |x-x_{0}|<\delta (\epsilon _{0})\!\,}$

Then, if ${\displaystyle |x-x_{0}|<\delta (\epsilon _{0})\!\,}$, by Triangle Inequality, we have

${\displaystyle \epsilon _{0}<|f(x_{0})|<|f(x)|+|f(x_{0})-f(x)|<|f(x)|+{\frac {\epsilon _{0}}{2}}\!\,}$

which implies

${\displaystyle {\frac {\epsilon _{0}}{2}}<|f(x)|\!\,}$,

whenever ${\displaystyle |x-x_{0}|<\delta (\epsilon _{0})\!\,}$

Let ${\displaystyle x_{0}\!\,}$ be a number greater than ${\displaystyle M\!\,}$. Note that ${\displaystyle \epsilon _{0}\!\,}$ and ${\displaystyle \delta (\epsilon _{0})\!\,}$ do not depend on ${\displaystyle M\!\,}$. With this in mind, note that

${\displaystyle \int _{[-M,M]^{C}}|f|>\int _{x_{0}}^{x_{0}+\delta (\epsilon _{0})}|f|>{\frac {\epsilon _{0}}{2}}\delta (\epsilon _{0})\qquad {\mbox{(*)}}\!\,}$

Then,

${\displaystyle 0=\lim _{M\rightarrow \infty }\int _{[-M,M]^{C}}|f|\geq {\frac {\epsilon _{0}}{2}}\delta (\epsilon _{0})\!\,}$

which is a huge contradiction.

Therefore,

${\displaystyle \lim _{|x|\rightarrow \infty }|f(x)|=0\!\,}$

Remark If we choose to work with the assumption that ${\displaystyle \lim _{x\rightarrow -\infty }|f|\neq 0\!\,}$ , then in (*), we just need to work with

${\displaystyle \int _{x_{0}-\delta (\epsilon _{0})}^{x_{0}}|f|\!\,}$

instead of the original one

By uniform continuity, for all ${\displaystyle \epsilon >0\!\,}$, there exists ${\displaystyle \delta (\epsilon )>0\!\,}$ such that for all ${\displaystyle x_{1},x_{2}\in R\!\,}$,

${\displaystyle |f(x_{1})-f(x_{2})|<\epsilon \!\,}$

if

${\displaystyle |x_{1}-x_{2}|<\delta (\epsilon )\!\,}$

Assume for the sake of contradiction there exists ${\displaystyle \epsilon _{0}>0\!\,}$ such that for all ${\displaystyle M>0\!\,}$, there exists ${\displaystyle x\in R\!\,}$ such that ${\displaystyle |x|>M\!\,}$ and ${\displaystyle |f(x)|\geq \epsilon _{0}\!\,}$.

Let ${\displaystyle M=1\!\,}$, then there exists ${\displaystyle x_{1}\!\,}$ such that ${\displaystyle |x_{1}|>M\!\,}$ and ${\displaystyle |f(x_{1})|\geq \epsilon _{0}\!\,}$.

Let ${\displaystyle M=|x_{1}|+3\delta (\epsilon _{0})\!\,}$, then there exists ${\displaystyle x_{2}\!\,}$ such that ${\displaystyle |x_{2}|>M\!\,}$ and ${\displaystyle |f(x_{2})|\geq \epsilon _{0}\!\,}$.

Let ${\displaystyle M=|x_{n}|+3\delta (\epsilon _{0})\!\,}$, then there exists ${\displaystyle x_{n+1}\!\,}$ such that ${\displaystyle |x_{n+1}|>M\!\,}$ and ${\displaystyle |f(x_{n+2})|\geq \epsilon _{0}\!\,}$.

So we have ${\displaystyle \{I_{n}\}=\{(x_{n}-\delta (\epsilon _{0}),x_{n}+\delta (\epsilon _{0})\}\!\,}$ with ${\displaystyle I_{i}\cap I_{j}=\emptyset \!\,}$ if ${\displaystyle i\neq j\!\,}$ and ${\displaystyle |f(x)|\geq \epsilon _{0}\!\,}$ for all ${\displaystyle x\in I_{n}\!\,}$ and for all ${\displaystyle n\!\,}$.

In other words, we are choosing disjoint subintervals of the real line that are of length ${\displaystyle 2\delta (\epsilon _{0})\!\,}$, centered around each ${\displaystyle x_{i}\!\,}$ for ${\displaystyle i=1,2,3,\ldots \!\,}$, and separated by at least ${\displaystyle \delta (\epsilon _{0})\!\,}$.

Hence,

${\displaystyle {\begin{aligned}\int _{R}|f(x)|dx&\geq \sum _{n=1}^{\infty }\int _{I_{n}}|f(x)|dx\\&\geq \sum _{n=1}^{\infty }\epsilon _{0}\int _{I_{n}}dx\\&=\sum _{n=1}^{\infty }\epsilon _{0}\cdot 2\delta (\epsilon _{0})\\&=+\infty \end{aligned}}}$

which contradicts the assumption that ${\displaystyle f\in L^{1}(R)\!\,}$.

Therefore, for all ${\displaystyle \epsilon >0\!\,}$ there exists ${\displaystyle M>0\!\,}$ such that for all ${\displaystyle |x|>M\!\,}$,

${\displaystyle |f(x)|<\epsilon \!\,}$

i.e.

${\displaystyle \lim _{|x|\rightarrow \infty }f(x)=0\!\,}$

By absolute continuity, Fatou's Lemma, and hypothesis we have

${\displaystyle {\begin{aligned}\int _{R}|f^{\prime }(t)|dt&=\int _{R}{\underset {t\rightarrow 0^{+}}{\lim \inf }}\left|{\frac {f(x+t)-f(x)}{t}}\right|dx\\&\leq {\underset {t\rightarrow 0^{+}}{\lim \inf }}\int _{R}\left|{\frac {f(x+t)-f(x)}{t}}\right|dx\\&=0\end{aligned}}}$

Hence ${\displaystyle f^{\prime }(t)=0\!\,}$ a.e.

From the fundamental theorem of calculus, for all ${\displaystyle x\in R\!\,}$,

${\displaystyle f(x)-f(0)=\int _{0}^{x}f^{\prime }(t)dt=0}$

i.e. ${\displaystyle f(x)\!\,}$ is a constant ${\displaystyle c=f(0)\!\,}$.

Assume for the sake of contradiction that ${\displaystyle |c|>0\!\,}$, then

${\displaystyle \int _{R}|f(x)|dx=\int _{R}|c|dx=\infty \!\,}$.

which contradicts the hypothesis ${\displaystyle f\in L^{1}(R)\!\,}$. Hence,

${\displaystyle |c|=0\!\,}$

i.e. ${\displaystyle f(x)=0\!\,}$ for all ${\displaystyle x\in R\!\,}$

First, for all ${\displaystyle a,b\geq 0\!\,}$,

${\displaystyle {\begin{aligned}(a+b)&\leq (a+b)+2a^{\frac {1}{2}}b^{\frac {1}{2}}\\&=a+2a^{\frac {1}{2}}b^{\frac {1}{2}}+b\\&=(a^{\frac {1}{2}})^{2}+2a^{\frac {1}{2}}b^{\frac {1}{2}}+(b^{\frac {1}{2}})^{2}\\&=(a^{\frac {1}{2}}+b^{\frac {1}{2}})^{2}\end{aligned}}}$

Taking square roots of both sides of the inequality yields,

${\displaystyle (a+b)^{\frac {1}{2}}\leq a^{\frac {1}{2}}+b^{\frac {1}{2}}\!\,}$

Hence for all ${\displaystyle f,g\in L^{\frac {1}{2}}\!\,}$,

${\displaystyle {\begin{aligned}\int _{R}|af(x)+bg(x)|^{\frac {1}{2}}dx&\leq \int _{R}(|a|^{\frac {1}{2}}|f(x)|^{\frac {1}{2}}+|b|^{\frac {1}{2}}|g(x)|^{\frac {1}{2}})dx\\&=|a|^{\frac {1}{2}}\int _{R}|f(x)|^{\frac {1}{2}}dx+|b|^{\frac {1}{2}}\int _{R}|g(x)|^{\frac {1}{2}}dx\\&<\infty \end{aligned}}}$

Hence, ${\displaystyle L^{\frac {1}{2}}\!\,}$ is a linear space.

${\displaystyle (i)\!\,}$ Since ${\displaystyle d(f,g)\geq 0\!\,}$,

${\displaystyle d(f,g)=0\quad \Longleftrightarrow \quad f=g\,\,a.e.\!\,}$

${\displaystyle (ii)\!\,}$ Also, for all ${\displaystyle f,g,h\in L^{\frac {1}{2}}\!\,}$,

${\displaystyle {\begin{aligned}d(f,g)&=\int _{R}|f(x)-g(x)|^{\frac {1}{2}}dx\\&=\int _{R}(|f(x)-h(x)+h(x)-g(x)|^{\frac {1}{2}}dx\\&\leq \int _{R}(|f(x)-h(x)|+|h(x)-g(x)|)^{\frac {1}{2}}dx\\&\leq \int _{R}|f(x)-h(x)|^{\frac {1}{2}}dx+\int _{R}|h(x)-g(x)|^{\frac {1}{2}}dx\\&=d(f,h)+d(h,g)\end{aligned}}}$

From ${\displaystyle (i)\!\,}$ and ${\displaystyle (ii)\!\,}$ , we conclude that ${\displaystyle d(f,g)\!\,}$ is a metric space.

For ${\displaystyle a,b,c\geq 0\!\,}$,

${\displaystyle (a+b+c)^{\frac {1}{2}}\leq a^{\frac {1}{2}}+(b+c)^{\frac {1}{2}}\leq a^{\frac {1}{2}}+b^{\frac {1}{2}}+c^{\frac {1}{2}}\!\,}$

By induction, we then have for all ${\displaystyle a_{k}\geq 0\!\,}$ and all ${\displaystyle k\!\,}$

${\displaystyle \left(\sum _{k=1}^{n}a_{k}\right)^{\frac {1}{2}}\leq \sum _{k=1}^{n}a_{k}^{\frac {1}{2}}\!}$

We can equivalently prove completeness by showing that a subsequence of a Cauchy sequence converges.

If a subsequence of a Cauchy sequence converges, then the Cauchy sequence converges.

Choose ${\displaystyle \{f_{n}\}\!\,}$ such that for all ${\displaystyle n\!\,}$,

${\displaystyle d(f_{n},f_{n+1})<{\frac {1}{2^{n}}}\!\,}$

Rewrite ${\displaystyle f_{m}(x)\!\,}$ as a telescoping sum (successive terms cancel out) i.e.

${\displaystyle f_{m}(x)=f_{1}(x)+\underbrace {\sum _{n=1}^{m-1}(f_{n+1}(x)-f_{n}(x))} _{g_{m-1}(x)}\!\,}$.

The triangle inequality implies,

${\displaystyle |f_{m}(x)|^{\frac {1}{2}}\leq |f_{1}(x)|^{\frac {1}{2}}+|g_{m-1}(x)|^{\frac {1}{2}}\!\,}$

which means the sequence ${\displaystyle |f_{m}(x)|^{\frac {1}{2}}\!\,}$ is always dominated by the sequence on the right hand side of the inequality.

Let ${\displaystyle g_{m}(x)=\sum _{n=1}^{m}|f_{n+1}(x)-f_{n}(x)|}$, then

${\displaystyle g_{m}(x)\leq g_{m+1}(x)\!\,}$

and

${\displaystyle g_{m}(x)\geq 0\!\,}$.

In other words, ${\displaystyle \{g_{m}\}\!\,}$ is a sequence of increasing, non-negative functions. Note that ${\displaystyle g\!\,}$, the limit of ${\displaystyle \{g_{m}\}\!\,}$ as ${\displaystyle m\rightarrow \infty \!\,}$, exists since ${\displaystyle \{g_{m}\}\!\,}$ is increasing. (${\displaystyle g\!\,}$ is either a finite number ${\displaystyle L\!\,}$ or ${\displaystyle \infty \!\,}$.)

Also,

${\displaystyle {\begin{aligned}\int _{R}|g_{m}(x)|^{\frac {1}{2}}dx&=\int _{R}\sum _{n=1}^{m}|f_{n+1}(x)-f_{n}(x)|^{\frac {1}{2}}dx\\&=\sum _{n=1}^{m}\underbrace {\int _{R}|f_{n+1}(x)-f_{n}(x)|^{\frac {1}{2}}dx} _{d(f_{n+1},f_{n})}\\&\leq \sum _{n=1}^{m}{\frac {1}{2^{n}}}\\&\leq 1\end{aligned}}}$

Hence, for all ${\displaystyle m\!\,}$

${\displaystyle \int _{R}|g_{m}(x)|^{\frac {1}{2}}dx\leq 1\!\,}$

By the Monotone Convergence Theorem,

${\displaystyle {\begin{aligned}\int _{R}\lim _{m\rightarrow \infty }|g_{m}(x)|^{\frac {1}{2}}dx&=\lim _{m\rightarrow \infty }\int _{R}|g_{m}(x)|^{\frac {1}{2}}dx\\&\leq 1\\&<\infty \end{aligned}}}$

Hence,

${\displaystyle \lim _{m\rightarrow \infty }g_{m}(x)\in L^{\frac {1}{2}}\!\,}$

From the Lebesgue dominated convergence theorem,

${\displaystyle {\begin{aligned}\int _{R}\lim _{m\rightarrow \infty }|f_{m}|^{\frac {1}{2}}&=\lim _{m\rightarrow \infty }\int _{R}|f_{m}|^{\frac {1}{2}}\\&\leq \lim _{m\rightarrow \infty }\int _{R}|f_{1}(x)|^{\frac {1}{2}}+\int _{R}|g_{m-1}(x)|^{\frac {1}{2}}\\&<\infty \end{aligned}}}$

where the last step follows since ${\displaystyle f_{1},g_{m-1}\in L^{\frac {1}{2}}\!\,}$

Hence,

${\displaystyle \lim _{m\rightarrow \infty }f_{m}\in L^{\frac {1}{2}}\!\,}$

i.e. ${\displaystyle L^{\frac {1}{2}}\!\,}$ is complete.