# Tuesday, January 18


:::{.remark}
Last time:

- $\norm{f}_{\ell^p} = \qty{\sum \abs{f_n} ^p}^{1\over p}$
- $\norm{f}_{\ell^ \infty} = \sup_{n} \abs{f_n}$.

Today:

- $\ell_p = \ts{f \da (f_n) \st \norm{f}_{\ell^p} < \infty }$.
- Example: set $f^k \da (0, 0, \cdots, 1, 1, \cdots)$ with zeros for the first $k-1$ entires and ones for all remaining entries.
  Then $f^k_i \convergesto{k\to\infty}0$ for each fixed component at index $i$.
  So $f^k \to (0)$ component-wise, but $\norm{f_k}_{\ell^\infty} = 1$ for every $k$, so this doesn't converge in $\ell_\infty$.

- Recall the $\eps\dash\delta$ and limit definitions of continuity.
- Recall the definition of uniform continuity.
- For $\Omega \subseteq \RR^n$, write $C(\Omega)$ for the $\RR\dash$vector space of continuous bounded functionals $f: \Omega\to \CC$ with the norm $\norm{f}_{L^\infty} = \sup_{x\in \Omega} \abs{f(x)}$.
- Define $C^k(\Omega, \CC^m)$ to be functions with $k$ continuous partial derivatives which are bounded, and set $C^\infty(\Omega, \CC^m) = \intersect _{k\geq 0} C^k(\Omega, \CC^m)$. 
  Define a norm $\norm{f}_{C^k} = \sum_{j\leq k} \sup_{x\in \Omega} \abs{f^{(j)} (x)}$.
- Take $g(x) = 2-x^2$ and consider $\BB_{1\over 2}(g)$ in $C[0, 1]$ with $\norm{\wait}_{L^\infty}$.
- Show that convergent implies Cauchy-convergent using the triangle inequality.
- Lipschitz with $\abs{c} < 1$ implies Cauchy.
- Lemma 1.4.2: $\norm{f_{n+k} - f_n} \leq q^n (1-q)^{-1} \norm{f_1 - f_0}$ for all $k\geq 0$.
  Use
  \[
  \norm{f_{n+k} - f_n} = \norm{\sum_{j=1}^k (f_{n+j} - f_{n+j-1} ) } \leq \norm{f_1 - f_0} \sum_{j=1}^k q^{n+j-1} \convergesto{n\to\infty}0
  .\]
- Counterexample, not all normed spaces are complete: take $V = C[-1, 1]$ with $\norm{f}_{L^1} \da \int_{-1}^1 \abs{f(x) }\dx$.
  Define a sequence of functions $(f_n)$:

  <!-- Xournal file: /home/zack/SparkleShare/github.com/Notes/Class_Notes/2022/Spring/FunctionalAnalysis/sections/figures/2022-01-18_15-07.xoj -->

  ![](figures/2022-01-18_15-09-38.png)

  Check that $\norm{f_{n+1} - f_n}_{L^1} \leq q \norm{f_n - f_{n-1}}_{L^1}$, and pointwise $f_n \to \chi_{[-1, 0]}$ which is discontinuous and not in $C[-1, 1]$.

- Banach spaces: complete normed vector spaces.

- Series:
  - Convergent: $f \da \lim_N \sum_{n\leq N} f_n \in V$.
  - Absolute convergence: $\sum \norm{f_n} < \infty$.
  - In a Banach space, absolutely convergent series can be rearranged.

- Theorem: A normed space is complete iff absolute convergence $\implies$ convergence.
  Proof:
  - Step 1: show that every Cauchy sequence has a convergent subsequence.
  - Set $a_n \da \sup_{m > n} \norm{f_n - f_m}$, then Cauchy implies $a_n\to 0$ in $\RR$.
  - Get a convergent subsequence $a_{n_j} \leq j^{-2}$.
  - Set $g_j \da f_{n_j} - f_{n_{j+1}}$, then $g\da \sum g_j$ absolutely converges, say to $g$.
  - Note $f_{n_i} - f_{n_{i+1}} = \sum_{j=1}^i g_j$, so the subsequence $(f_{n_j})$ is convergent.
  - Step 2: use this to show that the original sequence $(f_n)$ converges.
  - Set $f = \lim f_n$, then $\norm{f_n - f} \leq \norm{f_n - f_{n_i}} + \norm{f_{n_i} - f}$, using Cauchy for the first $\eps$ and the convergent subsequence for the second.

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:::{.remark title="Some random notes"}
Some theorems that hold in Hilbert spaces but not necessarily Banach spaces:

![](figures/2022-01-18_14-48-04.png)

![](figures/2022-01-18_14-48-13.png)

![](figures/2022-01-18_14-54-38.png)

Absolute continuity:

![](figures/2022-01-18_14-55-21.png)

![](figures/2022-01-18_14-56-04.png)

$L_p^\loc$:

![](figures/2022-01-18_14-57-54.png)

$\ker L = 0$ may not be sufficient to guarantee bijectivity in infinite dimensions:

![](figures/2022-01-18_15-27-21.png)

Boundedness:

![](figures/2022-01-18_15-28-55.png)

Bounded iff continuous:


![](figures/2022-01-18_15-29-25.png)

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