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diff --git a/pages/unbounded-operators/adjoint-operators.md b/pages/unbounded-operators/adjoint-operators.md
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+---
+title: Adjoint Operators
+parent: Unbounded Operators
+nav_order: 1
+published: false
+description: >
+  The Hellinger–Toeplitz Theorem states that an everywhere-defined symmetric
+  operator on a Hilbert space is bounded. We give a proof using the Uniform
+  Boundedness Theorem. We give another proof using the Closed Graph Theorem.
+# spellchecker:dictionaries latex
+# spellchecker:words Hellinger Toeplitz innerp hilb Schwarz functionals enspace Riesz
+---
+
+# {{ page.title }}
+
diff --git a/pages/unbounded-operators/graph-and-closedness.md b/pages/unbounded-operators/graph-and-closedness.md
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+---
+title: Graph and Closedness
+parent: Unbounded Operators
+nav_order: 1
+description: >
+  The Hellinger–Toeplitz Theorem states that an everywhere-defined symmetric
+  operator on a Hilbert space is bounded. We give a proof using the Uniform
+  Boundedness Theorem. We give another proof using the Closed Graph Theorem.
+# spellchecker:dictionaries latex
+# spellchecker:words Hellinger Toeplitz innerp hilb Schwarz functionals enspace Riesz
+---
+
+# {{ page.title }}
+
+
+{: .definition-title }
+
+> Definition (Graph of an Operator)
+>
+> The *graph* of an operator $T$ in a Hilbert space $\hilb{H}$
+> is the set of all pairs $(x,y) \in \hilb{H}\times\hilb{H}$
+> where $x$ lies in the domain of $T$ and $y=Tx$.
diff --git a/pages/unbounded-operators/hellinger-toeplitz-theorem.md b/pages/unbounded-operators/hellinger-toeplitz-theorem.md
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+---
+title: Hellinger–Toeplitz Theorem
+parent: Unbounded Operators
+nav_order: 10
+description: >
+  The Hellinger–Toeplitz Theorem states that an everywhere-defined symmetric
+  operator on a Hilbert space is bounded. We give a proof using the Uniform
+  Boundedness Theorem. We give another proof using the Closed Graph Theorem.
+# cspell:words Hellinger Toeplitz Schwarz Riesz functionals
+---
+
+# {{ page.title }}
+
+Conventions:
+{: .mb-0 }
+
+- Hilbert spaces are complex.
+- The inner product is anti-linear in its first argument.
+- Operators are linear and possibly unbounded.
+
+Recall that an operator $T : D(T) \to \hilb{H}$ in a Hilbert space $\hilb{H}$
+is called *symmetric*, if is has the property
+
+$$
+\innerp{Tx}{y} = \innerp{x}{Ty} \quad \forall x,y \in D(T).
+$$
+
+{: .theorem-title }
+> Hellinger–Toeplitz theorem
+>
+> An everywhere-defined symmetric operator on a Hilbert space is bounded.
+
+Consequently, a symmetric Hilbert space operator
+that is (truly) unbounded
+cannot be defined everywhere.
+
+---
+
+## Proof using the Uniform Boundedness Theorem
+
+Assume that $T$ is not bounded.
+Then there exists a sequence $(x_n)$ of unit vectors in $\hilb{H}$
+such that $\norm{Tx_n} \to \infty$.
+Consider the sequence $(f_n)$ of linear functionals on $\hilb{H}$,
+defined by
+
+$$
+f_n(y) = \innerp{Tx_n}{y} = \innerp{x_n}{Ty} \quad y \in \hilb{H}.
+$$
+
+The second identity is due to the symmetry of $T$.
+Apply Cauchy-Schwarz to both expressions to obtain the inequalities
+
+$$
+\abs{f_n(y)} \le \norm{Tx_n} \norm{y}
+\quad \text{and} \quad
+\abs{f_n(y)} \le \norm{x_n} \norm{Ty}
+$$
+
+for each $n \in \NN$ and $y \in \hilb{H}$.
+The first inequality shows that the functionals $f_n$ are bounded.
+The second one shows that, for fixed $y$,
+the sequence $(\abs{f_n(y)})$ is bounded by $\norm{Ty}$,
+since $\norm{x_n} = 1$ for all $n$.
+By the [Uniform Boundedness Theorem]({% link
+pages/functional-analysis-basics/the-fundamental-four/uniform-boundedness-theorem.md
+%}), $(\norm{f_n})$ is a bounded sequence.
+One has
+
+$$
+\norm{Tx_n}^2 = \abs{f_n(Tx_n)} \le \norm{f_n} \norm{Tx_n} \quad n \in \NN.
+$$
+
+Divide by $\norm{Tx_n}$ (if nonzero)
+to obtain $\norm{Tx_n} \le \norm{f_n}$ for all but finitely many $n$.
+Thus $(\norm{Tx_n})$ is a bounded sequence,
+contradicting $\norm{Tx_n} \to \infty$.
+{{ site.qed }}
+
+---
+
+## Proof using the Closed Graph Theorem
+
+By the [Closed Graph Theorem]({% link
+pages/functional-analysis-basics/the-fundamental-four/closed-graph-theorem.md %}),
+it is sufficient to show that the graph of $T$ is closed.
+Let $(x_n)$ be a convergent sequence of vectors in $\hilb{H}$
+such that the image sequence $(Tx_n)$ converges as well.
+Naming the limits $x$ and $z$, respectively, we have
+
+$$
+x_n \to x
+\quad \text{and} \quad
+Tx_n \to z.
+$$
+
+Continuity of the inner product implies
+
+$$
+\innerp{x_n}{Ty} \to \innerp{x}{Ty}
+\quad \text{and} \quad
+\innerp{Tx_n}{y} \to \innerp{z}{y}
+$$
+
+for all $y \in \hilb{H}$.
+Since $T$ is symmetric,
+the first assertion can be rewritten as
+
+$$
+\innerp{Tx_n}{y} \to \innerp{Tx}{y}.
+$$
+
+A sequence of complex numbers has at most one limit,
+hence $\innerp{Tx}{y} = \innerp{z}{y}$ for all $y$.
+By the Riesz representation theorem, $Tx=z$.
+{{ site.qed }}
diff --git a/pages/unbounded-operators/index.md b/pages/unbounded-operators/index.md
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+++ b/pages/unbounded-operators/index.md
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+---
+title: Unbounded Operators
+nav_order: 4
+has_children: true
+---
+
+# {{ page.title }}
diff --git a/pages/unbounded-operators/quadratic-forms.md b/pages/unbounded-operators/quadratic-forms.md
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--- /dev/null
+++ b/pages/unbounded-operators/quadratic-forms.md
@@ -0,0 +1,23 @@
+---
+title: Quadratic Forms
+parent: Unbounded Operators
+nav_order: 5
+published: false
+description: >
+  The Hellinger–Toeplitz Theorem states that an everywhere-defined symmetric
+  operator on a Hilbert space is bounded. We give a proof using the Uniform
+  Boundedness Theorem. We give another proof using the Closed Graph Theorem.
+# spellchecker:dictionaries latex
+# spellchecker:words Hellinger Toeplitz innerp hilb Schwarz functionals enspace Riesz
+---
+
+# {{ page.title }}
+
+
+{: .definition-title }
+
+> Definition (Graph of an Operator)
+>
+> The *graph* of an operator $T$ in a Hilbert space $\hilb{H}$
+> is the set of all pairs $(x,y) \in \hilb{H}\times\hilb{H}$
+> where $x$ lies in the domain of $T$ and $y=Tx$.