# Tuesday, January 11

:::{.remark}
This course: solving $Lf=g$ for $L$ a linear operator, in analogy to solving $Ax=b$ in matrices.
References:

- Hutson-Pym-Cloud, *Applications of Functional Analysis and Operator Theory*
- Reed-Simon, *Methods of Modern Mathematical Physics*
- Brezis, *Functional Analysis, Sobolev Spaces, and PDEs*

:::

:::{.remark}
The issue when passing to infinite-dimensional vector spaces: the topology matters.
E.g. the closure of the unit ball is closed and bounded and thus compact in finite dimensions, but this may no longer be true in $\RR^\infty$ or $\CC^\infty$.
Recall that a Banach space is a complete normed space, and is further a Hilbert space if the norm is induced by an inner product.
See the textbook for a review of vector spaces, metric spaces, norms, and inner products.
:::

:::{.example title="?"}
Our first example of infinite dimensional vector spaces: sequence spaces $\ell$ with elements $f \da (f_1, f_2, \cdots )$ with each $f_i\in \RR$.
:::

:::{.remark}
Linear subspaces are subspaces that contain zero, as opposed to affine subspaces.
An example is $C_0([0, L]; \RR) \leq C([0, L]; \RR)$, the subspace of bounded continuous functionals on $[0, L]$ which vanish at the endpoints.
For any subset $S \subseteq V$, write $[S]$ or $\spanof S$ for the linear span of $S$: all finite linear combinations of elements in $S$.
:::

:::{.example title="?"}
Let $V = C([-1, 1])$ and $x_1\neq x_2\in [-1, 1]$, and set $M_i \da \ts{f\in V \st f(x_i) = 0}$.
Then $M_i \leq V$ is a linear subspace, and in fact $V = M_1 + M_2$ but $V\neq M_1 \oplus M_2$ since the zero function is in both subspaces.
:::

:::{.remark}
Limits of finite operators are compact.
The classical example: set $(A_N)_{i, i} = {1\over i}$, so $A_N = \diag\qty{{1}, {1\over 2}, {1\over 3}, \cdots, {1\over N}}$.
Then $\spec A_N = \ts{1\over n}_{n\leq N}$, but $A\da \lim_N A_N$ is an operator with $0\in \bar{\spec(A)}$ as an accumulation point.
Exercise: what is $\ker A$? 
Is it nontrivial?
:::

:::{.definition title="Convexity"}
A subset $S \subseteq V$ is **convex** iff 
\[
tf + (1-t)g \in S \qquad \forall f, g\in S,\quad \forall t\in (0, 1]
.\]
Equivalently,
\[
{af+bg\over a+b}\in S \qquad \forall f, g\in S,\quad \forall a,b\geq 0
\]
where not both of $a$ and $b$ are zero.
The **convex hull** of $S$ is the smallest convex set containing $S$.
:::

:::{.remark}
Recall Holder's inequality:
\[
\norm{fg}_1 \leq \norm{f}_p \cdot \norm{g}_q
,\]
Schwarz's inequality
\[
\abs{\inner{f}{g}} \leq \norm{f} \norm{g} \qquad \norm{f} \da \sqrt{\inner{f}{f}}
,\]
and Minkowski's inequality
\[
\norm{f+g}_p \leq \norm{f}_p + \norm{g}_p
.\]

A nice proof of Cauchy-Schwarz:


![](figures/2022-01-11_15-43-17.png)

![](figures/2022-01-11_15-43-24.png)

:::