Improve knowledge base

1 files changed, 97 insertions, 65 deletions

diff --git a/source/know/concept/hilbert-space/index.md b/source/know/concept/hilbert-space/index.md
index 42b9cb1..2a60896 100644
--- a/source/know/concept/hilbert-space/index.md
+++ b/source/know/concept/hilbert-space/index.md
@@ -18,22 +18,32 @@ is an abstract **vector space** with a notion of length and angle.
 An abstract **vector space** $$\mathbb{V}$$ is a generalization
 of the traditional concept of vectors as "arrows".
 It consists of a set of objects called **vectors**
-which support the following (familiar) operations:
+that support the following (familiar) operations:
 
-+ **Vector addition**: the sum of two vectors $$V$$ and $$W$$, denoted by $$V + W$$.
-+ **Scalar multiplication**: product of a vector $$V$$ with a scalar $$a$$, denoted by $$a V$$.
++ **Vector addition**:
+  the sum of two vectors $$V$$ and $$W$$, denoted by $$V + W$$.
++ **Scalar multiplication**:
+  product of a vector $$V$$ with a scalar $$a$$, denoted by $$a V$$.
 
-In addition, for a given $$\mathbb{V}$$ to qualify as a proper vector
-space, these operations must obey the following axioms:
+In addition, for a given $$\mathbb{V}$$ to qualify as a proper vector space,
+these operations must have the following (again familiar) properties:
 
-+ **Addition is associative**: $$U + (V + W) = (U + V) + W$$
-+ **Addition is commutative**: $$U + V = V + U$$
-+ **Addition has an identity**: there exists a $$\mathbf{0}$$ such that $$V + 0 = V$$
-+ **Addition has an inverse**: for every $$V$$ there exists $$-V$$ so that $$V + (-V) = 0$$
-+ **Multiplication is associative**: $$a (b V) = (a b) V$$
-+ **Multiplication has an identity**: There exists a $$1$$ such that $$1 V = V$$
-+ **Multiplication is distributive over scalars**: $$(a + b)V = aV + bV$$
-+ **Multiplication is distributive over vectors**: $$a (U + V) = a U + a V$$
++ **Addition is associative**:
+  $$U + (V + W) = (U + V) + W$$
++ **Addition is commutative**:
+  $$U + V = V + U$$
++ **Addition has an identity**:
+  there exists a $$\mathbf{0}$$ such that $$V + 0 = V$$
++ **Addition has an inverse**:
+  for every $$V$$ there exists $$-V$$ so that $$V + (-V) = 0$$
++ **Multiplication is associative**:
+  $$a (b V) = (a b) V$$
++ **Multiplication has an identity**:
+  There exists a $$1$$ such that $$1 V = V$$
++ **Multiplication is distributive over scalars**:
+  $$(a + b)V = aV + bV$$
++ **Multiplication is distributive over vectors**:
+  $$a (U + V) = a U + a V$$
 
 A set of $$N$$ vectors $$V_1, V_2, ..., V_N$$ is **linearly independent**
 if the only way to satisfy the following relation
@@ -46,25 +56,28 @@ $$\begin{aligned}
 In other words, these vectors cannot be expressed in terms of each other.
 Otherwise, they would be **linearly dependent**.
 
-A vector space $$\mathbb{V}$$ has **dimension** $$N$$
-if only up to $$N$$ of its vectors can be linearly indepedent.
+$$\mathbb{V}$$ has **dimension** $$N$$
+if only up to $$N$$ of its vectors can be linearly independent.
 All other vectors in $$\mathbb{V}$$ can then be written
 as a **linear combination** of these $$N$$ **basis vectors**.
 
-Let $$\vu{e}_1, ..., \vu{e}_N$$ be the basis vectors,
-then any vector $$V$$ in the same space can be **expanded**
-in the basis according to the unique weights $$v_n$$,
-known as the **components** of $$V$$ in that basis:
+Let $$\vu{e}_1, ..., \vu{e}_N$$ be a (generally not unique)
+valid set of basis vectors of $$\mathbb{V}$$,
+then any vector $$V$$ in that space can be **expanded**
+in that basis according to unique weights $$v_n$$,
+called the **components** of $$V$$ in that basis:
 
 $$\begin{aligned}
     V = \sum_{n = 1}^N v_n \vu{e}_n
 \end{aligned}$$
 
-Using these, the vector space operations can then be implemented as follows:
+Using these components,
+the operations of vector addition and scalar multiplication
+can then be implemented as follows:
 
 $$\begin{gathered}
     V = \sum_{n = 1} v_n \vu{e}_n
-    \quad
+    \qquad
     W = \sum_{n = 1} w_n \vu{e}_n
     \\
     \quad \implies \quad
@@ -73,18 +86,24 @@ $$\begin{gathered}
     a V = \sum_{n = 1}^N a v_n \vu{e}_n
 \end{gathered}$$
 
+It is straightforward to see that this implementation satisfies the properties above.
+
 
 
 ## Inner product
 
-A given vector space $$\mathbb{V}$$ can be promoted to a **Hilbert space** or **inner product space**
+A given vector space $$\mathbb{V}$$ can be promoted
+to a **Hilbert space** or **inner product space**
 if it supports an operation $$\Inprod{U}{V}$$ called the **inner product**,
 which takes two vectors and returns a scalar,
 and has the following properties:
 
-+ **Skew symmetry**: $$\Inprod{U}{V} = (\Inprod{V}{U})^*$$, where $${}^*$$ is the complex conjugate.
-+ **Positive semidefiniteness**: $$\Inprod{V}{V} \ge 0$$, and $$\Inprod{V}{V} = 0$$ if $$V = \mathbf{0}$$.
-+ **Linearity in second operand**: $$\Inprod{U}{(a V + b W)} = a \Inprod{U}{V} + b \Inprod{U}{W}$$.
++ **Skew symmetry**:
+  $$\Inprod{U}{V} = (\Inprod{V}{U})^*$$, where $${}^*$$ is the complex conjugate.
++ **Positive semidefiniteness**:
+  $$\Inprod{V}{V} \ge 0$$, and $$\Inprod{V}{V} = 0$$ if $$V = \mathbf{0}$$.
++ **Linearity in second operand**:
+  $$\Inprod{U}{(a V + b W)} = a \Inprod{U}{V} + b \Inprod{U}{W}$$.
 
 The inner product describes the lengths and angles of vectors,
 and in Euclidean space it is implemented by the dot product.
@@ -93,34 +112,39 @@ The **magnitude** or **norm** $$|V|$$ of a vector $$V$$ is given by
 $$|V| = \sqrt{\Inprod{V}{V}}$$ and represents the real positive length of $$V$$.
 A **unit vector** has a norm of 1.
 
-Two vectors $$U$$ and $$V$$ are **orthogonal** if their inner product
-$$\Inprod{U}{V} = 0$$. If in addition to being orthogonal, $$|U| = 1$$ and
-$$|V| = 1$$, then $$U$$ and $$V$$ are known as **orthonormal** vectors.
+Two vectors $$U$$ and $$V$$ are **orthogonal**
+if their inner product $$\Inprod{U}{V} = 0$$.
+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 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).
+Orthonormality is desirable for basis vectors,
+so if they are 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:
+The implementation of the inner product in terms of components and basis vectors
+is as follows, which can easily be shown to satisfy the properties above:
 
 $$\begin{gathered}
     V = \sum_{n = 1}^N v_n \vu{e}_n
-    \quad
+    \qquad
     W = \sum_{n = 1}^N w_n \vu{e}_n
     \\
     \quad \implies \quad
     \Inprod{V}{W} = \sum_{n = 1}^N \sum_{m = 1}^N v_n^* w_m \Inprod{\vu{e}_n}{\vu{e}_j}
 \end{gathered}$$
 
-If the basis vectors $$\vu{e}_1, ..., \vu{e}_N$$ are already
-orthonormal, this reduces to:
+If the basis vectors $$\vu{e}_1, ..., \vu{e}_N$$ are already orthonormal,
+this reduces to:
 
 $$\begin{aligned}
     \Inprod{V}{W} = \sum_{n = 1}^N v_n^* w_n
 \end{aligned}$$
 
-As it turns out, the components $$v_n$$ are given by the inner product
-with $$\vu{e}_n$$, where $$\delta_{nm}$$ is the Kronecker delta:
+This suggests a way to calculate the components $$v_n$$:
+taking the inner product of $$V$$ with a basis vector $$\vu{e}_n$$
+"picks out" the corresponding component $$v_n$$.
+Let $$\delta_{nm}$$ be the Kronecker delta:
 
 $$\begin{aligned}
     \Inprod{\vu{e}_n}{V} = \sum_{m = 1}^N \delta_{nm} v_m = v_n
@@ -134,40 +158,46 @@ As the dimensionality $$N$$ tends to infinity, things may or may not
 change significantly, depending on whether $$N$$ is **countably** or
 **uncountably** infinite.
 
-In the former case, not much changes: the infinitely many **discrete**
-basis vectors $$\vu{e}_n$$ can all still be made orthonormal as usual,
-and as before:
+In the former case, not much changes:
+the infinitely many **discrete** basis vectors $$\vu{e}_n$$
+can all still be made orthonormal as usual, and as before:
 
 $$\begin{aligned}
     V = \sum_{n = 1}^\infty v_n \vu{e}_n
 \end{aligned}$$
 
-A good example of such a countably-infinitely-dimensional basis are the
-solution eigenfunctions of a [Sturm-Liouville problem](/know/concept/sturm-liouville-theory/).
+A good example of such a countably-infinitely-dimensional basis
+are the solution eigenfunctions of
+a [Sturm-Liouville problem](/know/concept/sturm-liouville-theory/).
 
-However, if the dimensionality is uncountably infinite, the basis
-vectors are **continuous** and cannot be labeled by $$n$$. For example, all
-complex functions $$f(x)$$ defined for $$x \in [a, b]$$ which
-satisfy $$f(a) = f(b) = 0$$ form such a vector space.
-In this case $$f(x)$$ is expanded as follows, where $$x$$ is a basis vector:
+However, if the dimensionality is uncountably infinite,
+the basis vectors are **continuous** and cannot be labeled by $$n$$.
+For example, all complex functions $$f(x)$$ defined on the interval $$x \in [a, b]$$
+satisfying the boundary condition $$f(a) = f(b) = 0$$, form such a vector space.
+In this case, every value of $$f(x)$$ is the component of
+an abstract vector $$\Ket{f}$$ with respect to a basis vector $$\Ket{x}$$:
 
 $$\begin{aligned}
-    f(x) = \int_a^b \Inprod{x}{f} \dd{x}
+    f(x) = \Inprod{x}{f}
 \end{aligned}$$
 
-Similarly, the inner product $$\Inprod{f}{g}$$ must also be redefined as
-follows:
+The inner product $$\Inprod{f}{g}$$ must be redefined as follows,
+effectively turning the sum over a discrete basis
+into an integral over a continuous basis:
 
 $$\begin{aligned}
     \Inprod{f}{g} = \int_a^b f^*(x) \: g(x) \dd{x}
 \end{aligned}$$
 
-The concept of orthonormality must be also weakened. A finite function
-$$f(x)$$ can be normalized as usual, but the basis vectors $$x$$ themselves
-cannot, since each represents an infinitesimal section of the real line.
+The concept of orthonormality must be also weakened.
+A finite function $$f(x)$$ can be normalized as usual,
+but the basis vectors $$x$$ themselves cannot,
+since each represents an infinitesimal section of the real line.
+So how to proceed?
 
-The rationale in this case is that action of the identity operator $$\hat{I}$$ must
-be preserved, which is given here in [Dirac notation](/know/concept/dirac-notation/):
+The rationale in this case is that the action
+of the identity operator $$\hat{I}$$ must be preserved,
+which is given here in [Dirac notation](/know/concept/dirac-notation/):
 
 $$\begin{aligned}
     \hat{I} = \int_a^b \Ket{\xi} \Bra{\xi} \dd{\xi}
@@ -181,8 +211,9 @@ $$\begin{aligned}
     = \int_a^b \Inprod{x}{\xi} f(\xi) \dd{\xi}
 \end{aligned}$$
 
-Since we want the latter integral to reduce to $$f(x)$$, it is plain to see that
-$$\Inprod{x}{\xi}$$ can only 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 $$\Inprod{x}{\xi}$$ can only be
+a [Dirac delta function](/know/concept/dirac-delta-function/),
 i.e $$\Inprod{x}{\xi} = \delta(x - \xi)$$:
 
 $$\begin{aligned}
@@ -191,12 +222,13 @@ $$\begin{aligned}
     = f(x)
 \end{aligned}$$
 
-Consequently, $$\Inprod{x}{\xi} = 0$$ if $$x \neq \xi$$ as expected for an
-orthogonal set of vectors, but if $$x = \xi$$ the inner product
-$$\Inprod{x}{\xi}$$ is infinite, unlike earlier.
+Consequently, $$\Inprod{x}{\xi} = 0$$ if $$x \neq \xi$$
+as expected for an orthogonal set of vectors,
+but if $$x = \xi$$ then the inner product $$\Inprod{x}{\xi}$$ is infinite,
+unlike earlier.
 
-Technically, because the basis vectors $$x$$ cannot be normalized, they
-are not members of a Hilbert space, but rather of a superset called a
-**rigged Hilbert space**. Such vectors have no finite inner product with
-themselves, but do have one with all vectors from the actual Hilbert
-space.
+Technically, because the basis vectors $$x$$ cannot be normalized,
+they are not members of a Hilbert space,
+but rather of a superset called a **rigged Hilbert space**.
+Such vectors have no finite inner product with themselves,
+but do have one with all vectors from the actual Hilbert space.