# Notes - Analysis II HT23, Uniform convergence

### Flashcards

What does it mean for $(f _ n)$ to converge pointwise to $f : E \to \mathbb R$?

What does the sequence of functions $f _ n(x) = x^n$ where $f _ n : [0, 1] \to \mathbb{R}$ converge (pointwise) to?

If $f _ n$ is continuous for every $n$, and $f _ n$ converges pointwise to $f$, then is $f$ continuous?

Not necessarily.

Let $(f _ n)$ be a sequence of functions where $f _ n : E \to \mathbb R$. Then what does it mean for $(f _ n)$ to converge uniformly to $f$?

In general, if every $f _ n$ in $(f _ n)$ is continuous and $(f _ n)$ converges pointwise to $f$, then $f$ is not necessarily continuous. But what stronger condition allows you to say $f$ will definitely be continuous if $f _ n$ is?

Uniform convergence.

Uniform continuity and uniform convergence both are stronger conditions than ordinary continuity and ordinary pointwise convergence. What’s the similarity between them?

They require showing that some value of $\delta$ or $N$ works for all points.

What notation is used for a sequence of functions $(f _ n)$ converging *uniformly* to $f$?

*uniformly*to $f$?

Let $(f _ n)$ be a sequence of functions where $f _ n : E \to \mathbb R$.

The definition of uniform convergence to $f$,

\[\forall \varepsilon> 0 \colon \exists N \in \mathbb N \text{ s.t. } \forall n>N \colon \forall x \in E \colon |f_n(x) - f(x)| < \varepsilon\]
is equite complicated and hard to use in practice. Can you state the equivalent definition in terms of supremums?

$(f _ n)$ converges uniformly to $f$ if and only if for sufficiently large $n$, $\{ \vert f _ n(x) - f(x) \vert : x \in E\}$ is bounded, and

\[s_n = \sup_{x \in E} |f_n(x) - f(x)| \to 0 \text{ as } n \to \infty\]Can you state the Cauchy criterion for uniform convergence?

Suppose $f _ n \to E \to \mathbb R$, then $f _ n$ converges uniformly if and only if

\[\forall \varepsilon > 0 \text { } \exists N \in \mathbb N \text{ s.t. } \forall m > n > N \text{ } \forall x \in E \text{ } |f_n(x) - f_m(x)| < \varepsilon\]When is the Cauchy criterion for uniform convergence

\[\forall \varepsilon > 0 \text { } \exists N \in \mathbb N \text{ s.t. } \forall m > n > N \text{ } \forall x \in E \text{ } |f_n(x) - f_m(x)| < \varepsilon\]
useful?

When we don’t have an expression for the function the series is converging to, i.e. when defining functions via power series.

Can you state the (slightly simpler) Cauchy criterion for uniform convergence for a sum $f _ n = \sum _ {k=1}^n u _ k$ of functions?

Can you state Weierstrass’ M-test for determining if a sum of functions converges uniformly?

Suppose $u _ k : E \to \mathbb R$ and $M _ k \in \mathbb R$ where $M _ k$ are constants, $\forall x \in E$, $ \vert u _ k(x) \vert \le M _ k$, and $\sum _ {k=1}^\infty M _ k$ converges, then

\[\sum u_k\]converges uniformly.

What useful theorem relates uniform convergence and power series?

If $\sum c _ k x^k$ has radius of convergence $R$ and if $ \vert \rho \vert < R$, then $\sum c _ k x^k$ converges uniformly on $\{x : \vert x \vert \le \rho\}$.

There is a theorem that states

If $\sum c _ k x^k$ has radius of convergence $R$ and if $\rho < R$, then $\sum c _ k x^k$ converges uniformly on $\{x : \vert x \vert \le \rho\}$.

Why is the condition about $\rho$ required, rather than $ \vert x \vert < R$?

If $\sum c _ k x^k$ has radius of convergence $R$ and if $\rho < R$, then $\sum c _ k x^k$ converges uniformly on $\{x : \vert x \vert \le \rho\}$.

Otherwise you could get a one-sided limit at the boundary.

If a sum of differentiable functions converges uniformly to $f$, does that mean that $f$ is also differentiable, like is the case with continuity?

No.

When proving that power series converge uniformly within their radius of convergence, i.e.

If $\sum c _ k x^k$ has radius of convergence $R$ and if $\rho < R$, then $\sum c _ k x^k$ converges uniformly on $\{x : \vert x \vert \le \rho\}$.

you use the $M$-test. How do you define $M _ k$, and can you verify that $u _ k(x) \le M _ k$ and that $\sum M _ k$ converges?

If $\sum c _ k x^k$ has radius of convergence $R$ and if $\rho < R$, then $\sum c _ k x^k$ converges uniformly on $\{x : \vert x \vert \le \rho\}$.

Define $M _ k = \vert c _ k \rho^k \vert $. Note $u _ k(x) = \vert c _ k x^k \vert \le M _ k$, and since $\rho$ is within the radius of convergence, $\sum \vert c _ k \rho _ k \vert = \sum M _ k$ converges.

When proving Weierstrass’ M-test, i.e.

Suppose $u _ k : E \to \mathbb R$ and $M _ k \in \mathbb R$ where $M _ k$ are constants, $\forall x \in E$, $ \vert u _ k(x) \vert \le M _ k$, and $\sum _ {k=1}^\infty M _ k$ converges, then

\[\sum u _ k(x)\]
converges uniformly.

what do you note to get started that the rest of the proof follows from almost immediately?

Suppose $u _ k : E \to \mathbb R$ and $M _ k \in \mathbb R$ where $M _ k$ are constants, $\forall x \in E$, $ \vert u _ k(x) \vert \le M _ k$, and $\sum _ {k=1}^\infty M _ k$ converges, then

converges uniformly.

$\sum^{\infty}\ _ {k=1} M\ _ k$ converges implies the Cauchy criterion, i.e. that

\[|M\_{m+1} + \cdots + M\_{n}| = M\_{m+1} + \cdots + M\_n < \varepsilon\]Can you give an example of a power series that is uniformly convergent within any bounded interval inside the interval of convergence, but not uniformly convergent on the whole interval?

Consider

\[f_n(x) = \frac{nx}{n^2x^2 + 1}\]
one way to show that this isn’t uniformly convergent on $[0, 1]$ is to use derivatives. What could you quickly note instead?

### Proofs

Prove:

Let $(f _ n)$ be a sequence of continuous functions on $E$ which converges uniformly to $f$ on $E$. Then if each $f _ n$ is continuous at $p \in E$, then $f$ is continuous at $p$.

Todo.

Prove the equivalance between uniform convergence and the following supremum condition:

$(f _ n)$ converges uniformly to $f$ if and only if for sufficiently large $n$, $\{ \vert f _ n(x) - f(x) \vert : x \in E\}$ is bounded, and

\[s_n \coloneqq \sup_{x \in E} |f_n(x) - f(x)| \to 0 \text{ as } n \to \infty\]

Todo.

Prove the Cauchy criterion for uniform convergence:

Suppose $f _ n \to E \to \mathbb R$, then $f _ n$ converges uniformly if and only if

\[\forall \varepsilon > 0 \text { } \exists N \in \mathbb N \text{ s.t. } \forall m > n > N \text{ } \forall x \in E \text{ } |f_n(x) - f_m(x)| < \varepsilon\]

Todo.

Prove Weierstrass’ M-test:

Suppose $u _ k : E \to \mathbb R$ and $M _ k \in \mathbb R$ where $M _ k$ are constants, $\forall x \in E$, $ \vert u _ k(x) \vert \le M _ k$, and $\sum _ {k=1}^\infty M _ k$ converges, then

\[\sum u_k(x)\]
converges uniformly.

Todo.

Prove the uniform convergence of power series within their radius of convergence:

If $\sum c _ k x^k$ has radius of convergence $R$ and if $\rho < R$, then $\sum c _ k x^k$ converges uniformly on $\{x : \vert x \vert \le \rho\}$.

If $\sum c _ k x^k$ has radius of convergence $R$ and if $\rho < R$, then $\sum c _ k x^k$ converges uniformly on $\{x : \vert x \vert \le \rho\}$.

Todo.