# Thursday, January 13

:::{.remark}
My notes:

- $K \subseteq \mch$ is **complete** iff $K^\perp = 0$.
- Bessel: for $f\in \mch$ write $f_n \da \inner{f}{e_n} e_n$, then $\norm{(f_n)}_{\ell^2(\CC)} \leq \norm{f}_{\mch}^2$.
- Best estimate: for any other sequence $(c_n) \in \ell^2(\CC), \norm{f - \sum c_n e_n} \geq \norm{f - \sum f_n e_n}$.
- For $\ts{e_n}$ orthonormal, $(c_n) \in \ell^2(\CC) \iff \sum c_n e_n$ converges.
  If the series converges, it can be rearranged.
- Differentiating through an integral:

![](figures/2022-01-13_14-58-19.png)

- Parseval, Plancherel, and Fourier inversion:

![](figures/2022-01-13_15-02-34.png)

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:::{.remark}
Last time: any norm yields a metric: $d(f, g) \da \norm{f-g}$.

- Open/closed balls: $B_r(f) \da \ts{x\st \norm{f-x} < r}$.
- Bounded subsets: contained in some ball of finite radius.
- $\diam S = \inf_{r, f} \diam B_r(f)$ is the diameter of the smallest ball containing $S$.
- $d(f, S) \da \inf_{x\in S} \norm{f-x}$.
- $V = \RR^n$ with $\norm{f}_2^2 \da \sum_{k\leq n} f_i^2$, $\bar B_0(1)$ is a metric space but not a vector space.
- For $L^2$, there are unique least squares projections, but uniqueness may fail for $L^1$.
  - Counterexample: take a line $M = \ts{\tv{\alpha, \alpha}}$ in $\RR^2$ of angle $\pi/4$ with respect to the $x\dash$axis and consider $f\da \tv[0, 1]$.
    Then for $g\da (\alpha, \alpha)$, $\norm{f-g} = \abs{1-\alpha} + \abs{\alpha} \geq \abs{1-\alpha + \alpha} = 1$, and the minimizer occurs for *any* $\alpha \in [0, 1]$.
  - Similar issues may happen for $L^\infty$ -- but $L^1, L^\infty$ have sharper tails than $L^2$, so this can be useful e.g. in image problems.
- If limits of sequences $(f_n)$ exist, i.e. $\norm{f_n - f_m}\to 0$, then the limiting function $f_n\to f$ is unique by the triangle inequality.
- Example from last time: $\diag\qty{ 1, {1\over 2}, \cdots, {1\over n}} \to A$ a compact self-adjoint operator with $\spec A = \ts{1\over n}_{n\geq 0}$.
  - What is $\ker A$? Note that $0\in \sigma(A)$, where $\sigma(A)$ is the set where $(A-I\lambda)\inv$ is not defined.
    It turns out $\ker A = \ts{0}$.
- Defining closures of subsets: for $S \subseteq V$, say $f\in \bar{S}$ iff there exists a sequence of not necessarily distinct points $f_n \in S$ with $f_n\to f$.
  - Say $S_1 \subseteq S_2 \subseteq V$ is closed in $S_2$ iff $S_1 = C \intersect S_2$ for some $C$ closed in $V$.
    The closure of $S_1$ in $S_2$ is $\cl_V(S_1) \intersect S_2$.
- A set that is neither open nor closed: $X \da [a, b] \intersect \QQ$, and $\bd X = [a,b] \contains X$ is actually larger.
- Recall the little $\ell^p$ norms: $\norm{(f_n)}_{p} \da \qty{ \sum \abs{f_n}^p }^{1\over p}$ and $\norm{(f_n)}_{\infty} \da \sup_n \abs{f_n}$.
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:::{.exercise title="?"}
- Prove Jensen's inequality for concave functions.
- Prove Young's inequality.
- Prove Holder's inequality.
  - Idea: consider $a = \hat{f_n} \da \abs{f_n}/\norm{f_n}^p, b = \hat{g_n} \da \abs{g_n}/\norm{g_n}_q$ and apply Young's after summing over $n$.
- Prove Minkowski's inequality.
  - Idea: use that $(p-1)q=p$ and apply the triangle inequality and then Holder to $\sum \abs{f_n + g_n}^p$.
    Also use that $q\inv = 1-p\inv$, and divide through this inequality at the end.
    Be sure to check the cases $\norm{f+g}_p = 0, \infty$.
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