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--- hamburgminicourse2017:why_hilbert_space [2017/04/02 15:54] – markus
+++ hamburgminicourse2017:why_hilbert_space [2017/04/03 16:11] (current) – markus
@@ Line 11: / Line 11: @@
 \]
-(Here the $x_n$ in the sum term can be the coordinates of the eigenstates of $H$ as an infinite matrix realtive to some basis, represented as a sequence $\mathbb{N} \rightarrow \mathbb{C}$.)
+(Here the $x_n$ in the sum term can be the coordinates of the eigenstates of $H$ as an infinite matrix relative to some basis, represented as a sequence $\mathbb{N} \rightarrow \mathbb{C}$.)
 The **Riesz--Fischer theorem** (1907) then told von Neumann, that the spaces of all such (normalized) objects are //isomorphic// as Hilbert spaces.
@@ Line 17: / Line 17: @@
 L^2(\Omega) \simeq \ell^2(\mathbb{N})
 \]
-e.g., $L^2(\mathbb{R}/2\pi) \simeq \ell^2(\mathbb{Z})$ with Fourier series or with any other orthonormal basis $(e_i)_{i \in I}$ of $L^2(\Omega)$:
+e.g., $L^2(\mathbb{R}/2\pi) \simeq \ell^2(\mathbb{Z})$ with Fourier series or with any other orthonormal basis $(e_i)_{i \in \mathbb{N}}$ of $L^2(\Omega)$:
 \[
-\psi \quad\longleftrightarrow\quad (\langle \psi,e_i \rangle)_{i \in I}.
+\psi \quad\longleftrightarrow\quad (\langle \psi,e_i \rangle)_{i \in \mathbb{N}}.
 \]
-The isomorphy holds on the level of //state spaces// but not on the level of //configuration spaces//, i.e. $\mathbb{R}^3$ and $\mathbb{N}$ (the numbering of orbitals). This explains the focus of QM on the Hilbert space (state space) instead of configuration space.
+The isomorphy holds on the level of //state spaces// but not on the level of //configuration spaces//, e.g. $\Omega=\mathbb{R}^3$ and $\mathbb{N}$ (the numbering of orbitals). This explains the focus of QM on the Hilbert space (state space) instead of configuration space.
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