# Tuesday, February 08

:::{.remark}
Motivating question: when is an operator equation solvable?
Today: relation between boundedness and continuity for linear operators.
Nonlinear operators next week.

- A map of vector spaces $V\to W$ is a linear map defined on some domain $D(A) \subseteq V$, where $D(A)$ need not equal $V$.
  - Notation: $A(f) = Af = g$.
  - $Af \subseteq W$ is the image of $f$, and $R(A) \da \ts{Af\st f\in D(A)} \subseteq W$ is the range.
  Preimages of $S \subseteq W$: $A\inv(S) = \ts{f\st f\in D(A) \text{ and } Af\in S}$.
- We distinguish operators with different domains, e.g. $Af \da f'$ can be the formula for distinct operators $A_1, A_2$ where $D(A_1) = C^\infty[0, 1] \subseteq C^0[0, 1]$ or $D(A_2) = C^1[0, 1] \subseteq C^0[0, 1]$, so $A_1\neq A_2$.
  - I.e. $A_1 = A_2 \iff D(A_1) = D(A_2)$ and $A_1 f = A_2 f$ for all $f\in D(A_1)=D(A_2)$.
  - If $A_1 f = A_2 f$ with $D(A_1) \subseteq D(A_2)$, say $A_2$ is an extension of $A_1$.
  The extension is proper iff $D(A_1)\neq D(A_2)$.
- Example operator: the Laplace equation $\laplacian f= g$ where $\laplacian = \del_{xx} + \del_{yy}$.
  We can take domains $g\in C[0, 1], L^2[0, 1], H^2[0, 1] = \ts{f\in L^2(0, 1) \st \del_{xx} f, \del_{yy} f\in L^2(0, 1) }$.
  - Why domains matter: boundary conditions affect what eigenfunctions you get. Examples where $A_1\neq A_2$:
  - Dirichlet boundary conditions: $\laplacian f = g, \ro{f}{\bd \Omega} = 0$.
    The relevant solution spaces is $D(A_1) = \ts{\phi\in C^2[0,1]^2 \st \ro \phi {\bd \Omega} = 0}$ for $A_1 \phi \da \laplacian \phi$.
  - Neumann boundary conditions: $\laplacian f = g, \ro{ \dd{f}{\vector n}}{\bd \Omega} = 0$, i.e. there is no flux across the boundary.
    The relevant solution space is $D(A_2) = \ts{\psi \in C^2[0,1]^2\st \ro{ \dd{\psi}{\vector n} }{\bd\Omega} = 0 }$ for $A_2 \psi \da \laplacian \psi$.

- Injectivity: for $A: V\to W$, for every $g\in R(A)$ there is exactly one $f\in D(A)$ with $Af=g$.
  Equivalently for linear operators, $Af = 0 \implies f=0$.
- Surjectivity: $R(A) = W$.
- Example: $A\da x\mapsto \sin(x)$ regarded as a function $A:\RR\to \RR$ is neither surjective nor injective: $R(A) = [-1, 1] \subsetneq \RR$, and $\sin(\pi \ZZ) = 0$.
- Linearity: for $Lf = g$, $L$ is linear if $L(af + bg) = aLf + bLg$.

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:::{.exercise title="?"}
Show that the following are equivalent conditions for continuity of $A: V\to W$ at $f_0\in D(A)$:

- $\norm{Af - Af_0}_W < \eps$ for all $f\in D(A)$ with $\norm{f-f_0}_V < \delta$
- For every sequence $\ts{f_k} \to f_0$, $Af_k \to Af_0$.
- Preimages of open sets in $W$ are again open in $V$.

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