diff options
author | Justin Gassner <justin.gassner@mailbox.org> | 2024-02-14 07:24:38 +0100 |
---|---|---|
committer | Justin Gassner <justin.gassner@mailbox.org> | 2024-02-14 07:24:38 +0100 |
commit | 28407333ffceca9b99fae721c30e8ae146a863da (patch) | |
tree | 67fa2b79d5c48b50d4e394858af79c88c1447e51 /pages/unbounded-operators/hellinger-toeplitz-theorem.md | |
parent | 777f9d3fd8caf56e6bc6999a4b05379307d0733f (diff) | |
download | site-28407333ffceca9b99fae721c30e8ae146a863da.tar.zst |
Update
Diffstat (limited to 'pages/unbounded-operators/hellinger-toeplitz-theorem.md')
-rw-r--r-- | pages/unbounded-operators/hellinger-toeplitz-theorem.md | 10 |
1 files changed, 3 insertions, 7 deletions
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 }} |