diff options
Diffstat (limited to 'source/know/concept/hilbert-space')
| -rw-r--r-- | source/know/concept/hilbert-space/index.md | 55 |
1 files changed, 30 insertions, 25 deletions
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 |