diff options
Diffstat (limited to 'pages/unbounded-operators')
| -rw-r--r-- | pages/unbounded-operators/adjoint-operators.md | 2 | ||||
| -rw-r--r-- | pages/unbounded-operators/graph-and-closedness.md | 15 | ||||
| -rw-r--r-- | pages/unbounded-operators/hellinger-toeplitz-theorem.md | 10 | ||||
| -rw-r--r-- | pages/unbounded-operators/quadratic-forms.md | 15 |
4 files changed, 13 insertions, 29 deletions
diff --git a/pages/unbounded-operators/adjoint-operators.md b/pages/unbounded-operators/adjoint-operators.md index a93e6d4..1b9fb0c 100644 --- a/pages/unbounded-operators/adjoint-operators.md +++ b/pages/unbounded-operators/adjoint-operators.md @@ -7,8 +7,6 @@ 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 index a9bf738..04a6789 100644 --- a/pages/unbounded-operators/graph-and-closedness.md +++ b/pages/unbounded-operators/graph-and-closedness.md @@ -6,17 +6,12 @@ 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 $y=Tx$. +{% 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 $y=Tx$. +{% enddefinition %} diff --git a/pages/unbounded-operators/hellinger-toeplitz-theorem.md b/pages/unbounded-operators/hellinger-toeplitz-theorem.md index 07d6b81..fea54be 100644 --- a/pages/unbounded-operators/hellinger-toeplitz-theorem.md +++ b/pages/unbounded-operators/hellinger-toeplitz-theorem.md @@ -6,7 +6,6 @@ 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 }} @@ -25,10 +24,9 @@ $$ \innerp{Tx}{y} = \innerp{x}{Ty} \quad \forall x,y \in D(T). $$ -{: .theorem-title } -> Hellinger–Toeplitz theorem -> -> An everywhere-defined symmetric operator on a Hilbert space is bounded. +{% theorem * Hellinger–Toeplitz Theorem %} +An everywhere-defined symmetric operator on a Hilbert space is bounded. +{% endtheorem %} Consequently, a symmetric Hilbert space operator that is (truly) unbounded @@ -75,7 +73,6 @@ 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 }} --- @@ -113,4 +110,3 @@ $$ 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/quadratic-forms.md b/pages/unbounded-operators/quadratic-forms.md index 5831b88..cb1c44a 100644 --- a/pages/unbounded-operators/quadratic-forms.md +++ b/pages/unbounded-operators/quadratic-forms.md @@ -7,17 +7,12 @@ 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 $y=Tx$. +{% 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 $y=Tx$. +{% enddefinition %} |