From 3138ead6bfd6e88e8cdbf9e4c32df64e18bc4595 Mon Sep 17 00:00:00 2001
From: Prefetch
Date: Fri, 9 Jun 2023 19:52:54 +0200
Subject: Improve knowledge base

---
 source/know/concept/hilbert-space/index.md | 55 ++++++++++++++++--------------
 1 file changed, 30 insertions(+), 25 deletions(-)

(limited to 'source/know/concept/hilbert-space/index.md')

diff --git a/source/know/concept/hilbert-space/index.md b/source/know/concept/hilbert-space/index.md
index 57926ce..42b9cb1 100644
--- a/source/know/concept/hilbert-space/index.md
+++ b/source/know/concept/hilbert-space/index.md
@@ -8,19 +8,20 @@ categories:
 layout: "concept"
 ---
 
-A **Hilbert space**, also called an **inner product space**, is an
-abstract **vector space** with a notion of length and angle.
+A **Hilbert space**, also called an **inner product space**,
+is an abstract **vector space** with a notion of length and angle.
+
 
 
 ## Vector space
 
-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:
+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:
 
-+ **Vector addition**: the sum of two vectors $$V$$ and $$W$$, denoted $$V + W$$.
-+ **Scalar multiplication**: product of a vector $$V$$ with a scalar $$a$$, denoted $$a V$$.
++ **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:
@@ -34,24 +35,26 @@ space, these operations must obey the following axioms:
 + **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 is to set all the scalar coefficients $$a_n = 0$$:
+A set of $$N$$ vectors $$V_1, V_2, ..., V_N$$ is **linearly independent**
+if the only way to satisfy the following relation
+is to set all the scalar coefficients $$a_n = 0$$:
 
 $$\begin{aligned}
     \mathbf{0} = \sum_{n = 1}^N a_n V_n
 \end{aligned}$$
 
-In other words, these vectors cannot be expressed in terms of each
-other. Otherwise, they would be **linearly dependent**.
+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. All other vectors in
-$$\mathbb{V}$$ can then be written as a **linear combination** of these $$N$$ **basis vectors**.
+A vector space $$\mathbb{V}$$ has **dimension** $$N$$
+if only up to $$N$$ of its vectors can be linearly indepedent.
+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 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:
 
 $$\begin{aligned}
     V = \sum_{n = 1}^N v_n \vu{e}_n
@@ -71,19 +74,20 @@ $$\begin{gathered}
 \end{gathered}$$
 
 
+
 ## Inner product
 
-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:
+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**: $$\Inprod{U}{(a V + b W)} = a \Inprod{U}{V} + b \Inprod{U}{W}$$.
 
-The inner product describes the lengths and angles of vectors, and in
-Euclidean space it is implemented by the dot product.
+The inner product describes the lengths and angles of vectors,
+and in Euclidean space it is implemented by the dot product.
 
 The **magnitude** or **norm** $$|V|$$ of a vector $$V$$ is given by
 $$|V| = \sqrt{\Inprod{V}{V}}$$ and represents the real positive length of $$V$$.
@@ -123,6 +127,7 @@ $$\begin{aligned}
 \end{aligned}$$
 
 
+
 ## Infinite dimensions
 
 As the dimensionality $$N$$ tends to infinity, things may or may not
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