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Diffstat (limited to 'latex/know/concept/hilbert-space/source.md')
| -rw-r--r-- | latex/know/concept/hilbert-space/source.md | 8 |
1 files changed, 4 insertions, 4 deletions
diff --git a/latex/know/concept/hilbert-space/source.md b/latex/know/concept/hilbert-space/source.md index 3f6ceb5..780fc0a 100644 --- a/latex/know/concept/hilbert-space/source.md +++ b/latex/know/concept/hilbert-space/source.md @@ -89,8 +89,8 @@ $\braket{U}{V} = 0$. If in addition to being orthogonal, $|U| = 1$ and $|V| = 1$, then $U$ and $V$ are known as **orthonormal** vectors. Orthonormality is desirable for basis vectors, so if they are -not already orthonormal, it is common to manually derive a new -orthonormal basis from them using e.g. the [Gram-Schmidt method](/know/concept/gram-schmidt-method). +not already like that, it is common to manually turn them into a new +orthonormal basis using e.g. the [Gram-Schmidt method](/know/concept/gram-schmidt-method). As for the implementation of the inner product, it is given by: @@ -171,8 +171,8 @@ $$\begin{aligned} = \int_a^b \braket{x}{\xi} f(\xi) \dd{\xi} \end{aligned}$$ -For the latter integral to turn into $f(x)$, it is plain to see that -$\braket{x}{\xi}$ must be a [Dirac delta function](/know/concept/dirac-delta-function/), +Since we want the latter integral to reduce to $f(x)$, it is plain to see that +$\braket{x}{\xi}$ can only be a [Dirac delta function](/know/concept/dirac-delta-function/), i.e $\braket{x}{\xi} = \delta(x - \xi)$: $$\begin{aligned} |