Vector Space

Algebraic View

（代数学方法卷一：基础架构）

Algebraic Structure Ladder (Set → Field)

Set $\overset{Op、结合性、幺元}{\to}$ Monoid $\overset{可逆}{\to}$ Group $\overset{交换、+、0}{\to}$ Additive Group $\overset{乘法半群+分配律}{\to}$ Rng $\overset{1}{\to}$ Ring $\overset{乘法交换}{\to}$ Commutative Ring $\overset{无零因子 / 乘法可消去}{\to}$ Integral Domain $\overset{乘法可逆}{\to}$ Field

Even more succinctly: a field is a commutative ring where 0 ≠ 1 and all nonzero elements are invertible under multiplication.

Additive Group $\overset{（环）R的纯量乘法（分配律、相容性、幺元）}{\to}$ Left/Right $R$ -module $M$ $\overset{除环D}{\to}$ $D$ -Vector Space $\overset{域K}{\to}$ $K$ -Vector Space, linear maps between each pair are $K$ -Module Homomorphisms

相容性缺失所成的反例见：Does a vector space need the compatibility of scalar multiplication with field multiplication axiom?

此处“数乘”为

\begin{matrix} (\cdot, \cdot) : C \times C^{2} \to C^{2}, \\ (a + b i, (c, d)) \mapsto (a c, a d) \end{matrix}

从形式观点看，模是带有一族乘法算子的加法群。
任意环对自身的左乘法构成左 $R$ -模，对右乘法构成右 $R$ -模。
交换环 $R$ 上的模不分左右，可以简称为 $R$ -模。
设 $D$ 为除环，我们称右 $D$ -模为 $D$ -向量空间。其子模，商模等也称为子空间，商空间。定义中选取左模或右模其实无关宏旨，当 $D$ 为域时更可以不论左右。

Analytic View

（Analysis I & 香蕉空间）

Vector Space Ladder (Ab → Hilb)

Additive Group $\overset{（环）R的纯量乘法（分配律、相容性、幺元）}{\to}$ Left/Right $R$ -module $M$ $\overset{除环D}{\to}$ $D$ -Vector Space $\overset{域K}{\to}$ $K$ -Vector Space $\overset{K = R 或 C}{\to}$ $K$ -Vector Space $X$ $\overset{范数（正定、正齐次/半范数、三角不等式/次可加）}{\to}$ Normed Vector Space $(X, ‖ \cdot ‖)$ $\overset{度量（正定、对称、三角不等式/次可加）}{\to}$ $(X, ‖ \cdot ‖)$ with induced metric $d$ , therefore a metric space $(X, d)$
① $\overset{内积（半双线性、共轭对称、正定）}{\to}$ Inner Product Space $(X, (\cdot | \cdot))$ $\overset{度量完备}{\to}$ Hilbert Space
② $\overset{度量完备}{\to}$ Banach Space $\overset{内积}{\to}$ Hilbert Space

Banach Space := Complete Normed Vector Space

Hilbert Space := Complete Inner Product Space

尽管形式上如此堆叠，实际诱导先后为 $Inner product ⟹ Norm ⟹ Metric$

Categorical View

选定一个域 𝕜, 令 $Vect (k)$ 为 𝕜 上所有向量空间构成的范畴, 态射为线性映射.

类此定义有限维向量空间范畴 ${Vect}_{f} (k)$ , 它是 $Vect (k)$ 的全子范畴.
对于任意 $k$ -向量空间 $V$ , 定义其对偶空间

V^{\lor} := {Hom}_{k} (V, k) = {k - 线性映射 V \to k} .

任一线性映射 $f : V_{1} \to V_{2}$ 诱导对偶空间的反向映射

\begin{aligned} f^{\lor} : V_{2}^{\lor} & ⟶ V_{1}^{\lor}, \\ [λ : V_{2} \to k] & ⟼ λ \circ f . \end{aligned}

易见 $D : V \to V^{\lor}$ , $f \mapsto f^{\lor}$ 定义了函子 $D : Vect (k)^{op} \to Vect (k)$ , 可以验证 $D$ 是忠实的. 根据注记 \ref{rem:op-functor}, 我们有合成函子 $D D^{op} : Vect (k) \to Vect (k)$ .

将 $D$ 限制于有限维向量空间, 便得到函子 $D : {Vect}_{f} (k)^{op} \to {Vect}_{f} (k)$ 和 $D D^{op} : {Vect}_{f} (k) \to {Vect}_{f} (k)$ . 分别称为对偶和双对偶函子.

考虑群范畴 $Grp$ . 对于任一个群 $G$ , 总是可以忘掉 $G$ 的群结构而视之为集合, 群同态当然也可以视为集合间的映射, 此程序给出\emph{忘却函子} $Grp \to Set$ . 准此要领可对其他结构定义忘却函子, 例如 $Top \to Set$ (忘掉空间的拓扑结构), $Vect (k) \to Ab$ (忘掉 $k$ -向量空间 $V$ 的纯量乘法, 只看它的加法群 $(V, +)$ , 这里 $k$ 是任意域) 等等, 不一一列举. 这类函子显然忠实而非全.

对于任意向量空间 $V$ 皆有求值映射

\begin{aligned} ev : V & ⟶ D D^{op} V = (V^{\lor})^{\lor} \\ v & ⟼ [λ \mapsto λ (v)] . \end{aligned}

对于任意线性映射 $f : V \to W$ , 从 $f^{\lor}$ 的定义不难检查以下图表

是交换的, 于是有 $e v : i d \to D D^{o p}$ . 容易看出 $e v : V \to D D^{o p} V$ 总是单射, 事实上可以
证明 ev 是双射当且仅当 $V$ 有限维. 一切限制到全子范畴 ${Vect}_{f} (k)$ 上, 遂有同构

ev : {id}_{{Vect}_{f} (k)} \tilde{\to} D D^{op} .

同一式子在相反范畴中诠释，便是

{id}_{{Vect}_{f} (k)^{op}} \tilde{\to} D^{op} D .

故函子 $D : {Vect}_{f} (k)^{op} \to {Vect}_{f} (k)$ 是范畴间的等价, 而 $D^{op} : {Vect}_{f} (k) \to {Vect}_{f} (k)^{op}$ 则是它的拟逆.

选定域 $k$ , 定义范畴 $Mat$ 如下: 其对象是 $Z_{\geq 0}$ , 对任意对象 $n, m \in Z_{\geq 0}$ , 定义 $Hom (n, m) := M_{m \times n} (k)$ 为域 $k$ 上的全体 $m \times n$ 矩阵 $A = (a_{i j})_{\begin{matrix} 1 \leq i \leq m \\ 1 \leq j \leq n \end{matrix}}$ 所成集合. 约定 $M_{0 \times n} (k) = M_{m \times 0} (k) := {0}$ . 态射的合成定义为寻常的矩阵乘法

\begin{aligned} Hom (n, m) \times Hom (m, k) & ⟶ Hom (n, k) \\ (A, B) & ⟼ B A . \end{aligned}

定义函子 $F : Mat \to {Vect}_{f} (k)$ 如下: 置 $F (n) = k^{\oplus n} := M_{n \times 1} (k)$ , 而对 $A \in Hom (n, m)$ , 线性映射 $F A : k^{\oplus n} \to k^{\oplus m}$ 是矩阵乘法 $v \mapsto A v$ . 我们断言 $F$ 是范畴等价.

这一切只是虚张声势的线性代数. 首先留意到 $F : Hom (n, m) \to {Hom}_{k} (k^{\oplus n}, k^{\oplus m})$ 是双射, 这无非是线性映射的矩阵表达. 再者, 从 $V ≃ k^{\oplus \dim V}$ ( $V$ 是 $k$ -向量空间) 可知 $F$ 是全忠实本质满的, 由定理 \ref{prop:functor-equiv-criterion} 可知它是范畴等价.

域 $k$ 上的向量空间范畴 $Vect (k)$ : 零空间是零对象, 零映射是零态射.

选定域 $k$ , 定义函子 $V : Set \to Vect (k)$ 如下: 对于集合 $X$ , 命 $V (X) := ⨁_{x \in X} k x$ 为以 $X$ 为基的 $k$ -向量空间. 任意映射 $f : X \to Y$ 皆诱导出线性映射 $V (f) : V (X) \to V (Y)$ , 它由在基上的限制 $f$ 所刻画. 令 $U : Vect (k) \to Set$ 为忘却函子, 则 $x \mapsto x \in V (X)$ 给出态射 $ι : X \to U V (X)$ . 尽管有些拗口, 不妨设想 $V (X)$ 是 $X$ 上的“自由向量空间”.

为阐明 $V (X)$ 的泛性质, 定义范畴 $(X / U)$ 使得其对象形如 $(W, i : X \to U (W))$ , 其中 $W \in Ob Vect (k)$ 而 $X \overset{i}{\to} U (W)$ 是 $Set$ 中的态射, 态射定为使下图交换的线性映射 $h : W_{1} \to W_{2}$ :

我们断言 $(V (X), ι)$ 是 $(X / U)$ 中的始对象. 这说的无非是对任意 $(W, i) \in Ob (X / U)$ , 存在唯一的 $h : V (X) \to W$ 使图表

交换. 由于 $X$ 是 $V (X)$ 的基, 这般 $h$ 是唯一确定了的.

上图涉及 $h$ 的条件称为 $V (X)$ 满足的泛性质. 根据命题 \ref{prop:initial-obj-uniqueness}, 我们说泛性质刻画了 $V (X)$ 连同 $ι : X \to U V (X)$ , 至多差一个唯一的同构. 之后我们还会遇到更精密的“自由对象”的构造, 如自由群.

Geometrical View

classical continuous geometries

Here we merely list, for future reference, several very classical geometries whose transformation groups are “continuous” rather than finite or discrete. We will not make the intuitively clear notion of continuous transformation group precise (this would involve defining the so-called topological groups or even Lie groups)

Finite-dimensional vector spaces

(V^{n} : GL (n))

n-dimensional orthonormal vector space

(V^{n} : O (n))

Affine spaces are, informally speaking, finite-dimensional vector spaces “without a fixed origin”. This means that their transformation groups $Aff (n)$ contain, besides $GL (n)$ , all parallel translations of the space (i.e., transformations of the space obtained by adding a fixed vector to all its elements).

(V^{n} : Aff (n)) or (V^{n} : Aff (n))

Euclidean spaces

(R^{n} : Sym (R^{n}))

Euclidean Geometry

Inclusive Relations Between Spaces

${inner product vector spaces} ⊊ {normed vector spaces} ⊊ {metric spaces} ⊊ {topological spaces} .$

Any metric space is a Hausdorff space.

"topology on $X$ induced from the metric $d$ "

"metric on $E$ induced from the norm $‖ \cdot ‖$ "

"norm induced from the scalar product $(\cdot | \cdot)$ "

Euclidean inner product: $(x | y) := \sum_{j = 1}^{m} x_{j} {\bar{y}}_{j}$

Hilbert norm: "norm $‖ x ‖ := \sqrt{(x | x)}$ on $E$ induced from the scalar product $(\cdot | \cdot)$ "

(Jordan-von Neumann theorem) keys are parallelogram law and polarization identity.

Euclidean norm: "norm $| x | := \sqrt{(x | x)} = \sqrt{\sum_{j = 1}^{m} | x_{j} |^{2}}$ on $E$ induced from the Euclidean inner product $(\cdot | \cdot)$ "

metric induced from the norm: $d (x, y) = ‖ x - y ‖$

拓扑空间

Normed vector space - Wikipedia

Every normed vector space can be "uniquely extended" to a Banach space, which makes normed spaces intimately related to Banach spaces. Every Banach space is a normed space but converse is not true. For example, the set of the finite sequences of real numbers can be normed with the Euclidean norm, but it is not complete for this norm.

Of special interest are complete normed spaces, which are known as Banach spaces. Every normed vector space $V$ sits as a dense subspace inside some Banach space; this Banach space is essentially uniquely defined by $V$ and is called the completion of $V$ .

Two norms on the same vector space are called equivalent if they define the same topology. On a finite-dimensional vector space (but not infinite-dimensional vector spaces), all norms are equivalent (although the resulting metric spaces need not be the same) And since any Euclidean space is complete, we can thus conclude that all finite-dimensional normed vector spaces are Banach spaces.

此处等价有类TCS中 $Θ$ ，见 Godement 书。

Normable spaces

Metrizable topological vector space

Norms

Taxicab/Manhattan norm: $| x |_{1}$
Euclidean norm: $| x |_{2}$
Chebyshev/uniform/supremum/infinity norm: $| x |_{\infty}$
$p$ -norm: $| x |_{p}$ 幂平均不等式

What does "all norms are equivalent" actually mean? : r/mathematics

general topology - Definition of Equivalent Norms - Mathematics Stack Exchange

ABCDEFGHIJKLMNOPQRSTUVWXYZ
$ABCDEFGHIJKLMNOPQRSTUVWXYZ$

Bilinear Functional

Sesquilinear form (in second argument**)

conjugate-linear (i.e. antilinear)

Hermitian bilinear functional (in Halmos book)

Quadratic Form

Polarization_identity

Antilinear in second argument

Inner product space

The article on Hilbert spaces has several examples of inner product spaces, wherein the metric induced by the inner product yields a complete metric space. An example of an inner product space which induces an incomplete metric is the space $C ([a, b])$ , of continuous complex valued functions $f$ and $g$ on the interval $[a, b]$ . The inner product is

⟨ f, g ⟩ = \int_{a}^{b} f (t) \overset{―}{g (t)} d t .

This space is not complete; consider for example, for the interval $[- 1, 1]$ the sequence of continuous "step" functions, ${f_{k}}_{k},$ defined by:

f_{k} (t) = {\begin{cases} 0 & t \in [- 1, 0] \\ 1 & t \in [\frac{1}{k}, 1] \\ k t & t \in (0, \frac{1}{k}) \end{cases}

This sequence is a Cauchy sequence for the norm induced by the preceding inner product, which does not converge to a continuous function.

Banach 空间

有限维向量空间上的范数总是完备的. 因此, 带有范数的有限维向量空间都是 Banach 空间. 但这种情形太简单, 人们更感兴趣的是无限维的 Banach 空间.

Banach space

Like all norms, this norm induces a translation invariant distance function, called the canonical or (norm) induced metric, defined for all vectors

Complete metric space

The space $R$ of real numbers and the space $C$ of complex numbers (with the metric given by the absolute difference) are complete, and so is Euclidean space $R^{n}$ , with the usual distance metric. In contrast, infinite-dimensional normed vector spaces may or may not be complete; those that are complete are Banach spaces. The space $C [a, b]$ of continuous real-valued functions on a closed and bounded interval is a Banach space, and so a complete metric space, with respect to the supremum norm. However, the supremum norm does not give a norm on the space $C (a, b)$ of continuous functions on $(a, b)$ , for it may contain unbounded functions. Instead, with the topology of compact convergence, $C (a, b)$ can be given the structure of a Fréchet space: a locally convex topological vector space whose topology can be induced by a complete translation-invariant metric.

Is the absolute value function a metric? - Mathematics Stack Exchange

$| x | = ∥ x ∥_{1} = ∥ x ∥_{2} = ∥ x ∥_{\infty}$

Neighbourhood

内有开球