线性空间 和度量空间 ，分别具有代数结构和拓扑结构；赋范线性空间 ；内积空间 ，作为赋范线性空间的特例；R n \small R^n R n
什么是空间？In mathematics, a space is a set with some added structure.(@Wikipedia)V \small V V V \small V V + : V × V → V \small +:V\times V \to V + : V × V → V ∀ x , y ∈ V \small \forall \;x,y \in V ∀ x , y ∈ V V \small V V γ \gamma γ x x x y y y 和 ，记为 x + y x+y x + y F \small F F ⋅ : F × V → V \small \cdot:F \times V \to V ⋅ : F × V → V ∀ a ∈ F , x ∈ V \small \forall \, a \in F,\,x \in V ∀ a ∈ F , x ∈ V V \small V V δ \delta δ a a a x x x 数量乘积 ，记为 a x ax a x
加法：
1. x + ( y + z ) = ( x + y ) + z 1.\,x+(y+z)=(x+y)+z 1 . x + ( y + z ) = ( x + y ) + z 2. x + y = y + x 2.\,x+y=y+x 2 . x + y = y + x 3. T h e r e e x i s t s a n e l e m e n t 3.\,There\,\,exists\,\,an\,\,element 3 . T h e r e e x i s t s a n e l e m e n t 0 \bf 0 0 ∈ V , s u c h t h a t x + ∈ V,such\,\,that\,\,x \,+ ∈ V , s u c h t h a t x + 0 \bf 0 0 = x , ∀ x ∈ V = x ,\forall \, x ∈ V = x , ∀ x ∈ V 4. ∀ x ∈ V , t h e r e e x i s t s a n e l e m e n t − x ∈ V , s u c h t h a t x + ( − x ) = 4.\,\forall \,x ∈ V, there \,\, exists \,\,an \,\,element \,\,−x ∈ V, such \,\,that\,\, x + (−x) = 4 . ∀ x ∈ V , t h e r e e x i s t s a n e l e m e n t − x ∈ V , s u c h t h a t x + ( − x ) = 0 \bf 0 0
数乘 ( a , b ∈ F ) \small (a,b \in F) ( a , b ∈ F )
5. 1 x = x 5.\,1x=x 5 . 1 x = x 6. ( a b ) x = a ( b x ) 6.\,(ab)x=a(bx) 6 . ( a b ) x = a ( b x )
加法和数乘：
7. ( a + b ) x = a x + b x 7.\,(a+b)x=ax+bx 7 . ( a + b ) x = a x + b x 8. a ( x + y ) = a x + a y 8.\,a(x+y)=ax+ay 8 . a ( x + y ) = a x + a y
如果这些条件都满足，则称 V \small V V F \small F F 线性空间 或向量空间 ，其中的元素称为向量 .
度量空间 = 集合 + 拓扑结构V \small V V V \small V V d : V × V → R \small d:V\times V\to R d : V × V → R ∀ x , y ∈ V \small \forall \;x,y \in V ∀ x , y ∈ V R \small R R δ \delta δ x , y x,y x , y 距离 ，记为 d ( x , y ) d(x,y) d ( x , y )
1. d ( x , y ) ≥ 0 , ∀ x , y ∈ V 1.\,d(x,y)\geq0,\forall \;x,y \in V\, 1 . d ( x , y ) ≥ 0 , ∀ x , y ∈ V d ( x , y ) = 0 ⇔ x = y \,\, d(x,y)=0 \Leftrightarrow x=y d ( x , y ) = 0 ⇔ x = y 2. d ( x , z ) ≤ d ( x , y ) + d ( y , z ) 2.\,d(x,z)\leq d(x,y)+d(y,z) 2 . d ( x , z ) ≤ d ( x , y ) + d ( y , z )
则称 ( V , d ) \small (V,d) ( V , d ) 度量空间 或距离空间 ，其中的元素称为点 .
赋范线性空间 = 线性空间 + 范数，即给线性空间穿上拓扑结构的外衣.V \small V V R \small R R ∥ ⋅ ∥ : V → R \small \Vert \cdot\Vert: V \to R ∥ ⋅ ∥ : V → R ∀ x ∈ V \small \forall \;x \in V ∀ x ∈ V R \small R R δ \delta δ x x x 范数 ，记为 ∥ x ∥ \small \Vert x\Vert ∥ x ∥
1. ∥ x ∥ ≥ 0 1.\,\Vert x\Vert \geq 0 \, 1 . ∥ x ∥ ≥ 0 ∥ x ∥ = 0 ⇔ x = 0 \,\Vert x\Vert=0 \Leftrightarrow x=0 ∥ x ∥ = 0 ⇔ x = 0 2. ∥ a x ∥ = ∣ a ∣ ∥ x ∥ , a ∈ R 2.\,\Vert ax\Vert=\vert a\vert \Vert x\Vert,\, a \in R 2 . ∥ a x ∥ = ∣ a ∣ ∥ x ∥ , a ∈ R 3. ∥ x + y ∥ ≤ ∥ x ∥ + ∥ y ∥ , x , y ∈ V 3.\,\Vert x+y\Vert \leq \Vert x\Vert + \Vert y\Vert,\, x,y \in V 3 . ∥ x + y ∥ ≤ ∥ x ∥ + ∥ y ∥ , x , y ∈ V
则称 ( V , ∥ ⋅ ∥ ) \small (V, \Vert \cdot\Vert) ( V , ∥ ⋅ ∥ ) 赋范线性空间 . d ( x , y ) = ∥ x − y ∥ \small d(x,y)=\Vert x-y\Vert d ( x , y ) = ∥ x − y ∥ 1. d ( x , y ) ≥ 0 , ∀ x , y ∈ V 1. \,\small d(x,y)\geq0,\forall \;x,y \in V\, 1 . d ( x , y ) ≥ 0 , ∀ x , y ∈ V d ( x , y ) = 0 ⇔ x = y \,\, d(x,y)=0 \Leftrightarrow x=y d ( x , y ) = 0 ⇔ x = y 2. d ( x , z ) = ∥ x − z ∥ = ∥ x − y + y − z ∥ ≤ ∥ x − y ∥ + ∥ y − z ∥ = d ( x , y ) + d ( y , z ) \begin{aligned}2. \,\small d(x,z)&=\Vert x-z\Vert=\Vert x-y+y-z\Vert\\&\leq \Vert x-y\Vert + \Vert y-z\Vert\\ &=d(x,y)+d(y,z)\end{aligned} 2 . d ( x , z ) = ∥ x − z ∥ = ∥ x − y + y − z ∥ ≤ ∥ x − y ∥ + ∥ y − z ∥ = d ( x , y ) + d ( y , z ) d d d V \small V V ( V , d ) \small (V,d) ( V , d ) V \small V V
内积空间 = 线性空间 + 内积V \small V V ( ⋅ , ⋅ ) : V × V → R \small (\,\cdot\,,\cdot\,): V \times V \to R ( ⋅ , ⋅ ) : V × V → R ∀ x , y ∈ V \small \forall \;x,y \in V ∀ x , y ∈ V R \small R R δ \delta δ x x x y y y 内积 ，记为 ( x , y ) (x,y) ( x , y )
1. ( x , x ) ≥ 0 1.\,(x,x)\geq0 1 . ( x , x ) ≥ 0 ( x , x ) = 0 ⇔ x = 0 \,\, (x,x)=0 \Leftrightarrow x=0 ( x , x ) = 0 ⇔ x = 0 2. ( x , y ) = ( y , x ) 2.\,(x,y)=(y,x) 2 . ( x , y ) = ( y , x ) 3. ( a x , z ) = a ( x , z ) , a ∈ R 3.\,(ax,z)=a(x,z),\,a \in R 3 . ( a x , z ) = a ( x , z ) , a ∈ R 4. ( x + y , z ) = ( x , z ) + ( y , z ) 4.\,(x+y,z)=(x,z)+(y,z) 4 . ( x + y , z ) = ( x , z ) + ( y , z )
则称 ( V , ( ⋅ , ⋅ ) ) \small (V,(\,\cdot\,,\cdot\,)) ( V , ( ⋅ , ⋅ ) ) 内积空间 . ∥ x ∥ = ( x , x ) \small \Vert x\Vert=\sqrt{(x,x)} ∥ x ∥ = ( x , x ) 1. ∥ x ∥ ≥ 0 1. \,\small \Vert x\Vert \geq 0 \, 1 . ∥ x ∥ ≥ 0 ∥ x ∥ = 0 ⇔ x = 0 \,\Vert x\Vert=0 \Leftrightarrow x=0 ∥ x ∥ = 0 ⇔ x = 0 2. ∥ a x ∥ = ( a x , a x ) = a 2 ( x , x ) = ∣ a ∣ ∥ x ∥ 2. \,\small \Vert ax\Vert=\sqrt{(ax,ax)}=\sqrt{a^2(x,x)}=\vert a\vert \Vert x\Vert 2 . ∥ a x ∥ = ( a x , a x ) = a 2 ( x , x ) = ∣ a ∣ ∥ x ∥ 3. ∥ x + y ∥ ≤ ∥ x ∥ + ∥ y ∥ 3. \,\small \Vert x+y\Vert \leq \Vert x\Vert + \Vert y\Vert 3 . ∥ x + y ∥ ≤ ∥ x ∥ + ∥ y ∥ 柯西-施瓦兹(Cauchy-Schwarz)不等式 ( x , y ) 2 ≤ ( x , x ) ( y , y ) (x,y) ^2\leq(x,x)(y,y) ( x , y ) 2 ≤ ( x , x ) ( y , y ) x = 0 x=0 x = 0 ( t x + y , t x + y ) = ( x , x ) t 2 + 2 ( x , y ) t + ( y , y ) ≥ 0 (tx+y,tx+y) = (x,x)t^2+2(x,y)t+(y,y)\geq0 ( t x + y , t x + y ) = ( x , x ) t 2 + 2 ( x , y ) t + ( y , y ) ≥ 0 t t t Δ = 4 ( x , y ) 2 − 4 ( x , x ) ( y , y ) ≤ 0 \Delta=4(x,y)^2-4(x,x)(y,y)\leq0 Δ = 4 ( x , y ) 2 − 4 ( x , x ) ( y , y ) ≤ 0 ( x , y ) 2 ≤ ( x , x ) ( y , y ) \small (x,y)^2\leq(x,x)(y,y) ( x , y ) 2 ≤ ( x , x ) ( y , y ) ( x , y ) ≤ ( x , x ) ( y , y ) \small (x,y)\leq \sqrt{(x,x)(y,y)} ( x , y ) ≤ ( x , x ) ( y , y ) t x + y = 0 tx+y=0 t x + y = 0 = = = ∥ x + y ∥ 2 = ( x + y , x + y ) = ( x , x ) + ( x , y ) + ( y , x ) + ( y , y ) ≤ ( x , x ) + 2 ( x , x ) ( y , y ) + ( y , y ) = ( ( x , x ) + ( y , y ) ) 2 = ( ∥ x ∥ + ∥ y ∥ ) 2 \begin{aligned}\small \Vert x+y\Vert^2&=(x+y,x+y)\\&=(x,x)+(x,y)+(y,x)+(y,y)\\ &\leq (x,x)+2\sqrt{(x,x)(y,y)}+(y,y)\\&=(\sqrt{(x,x)}+\sqrt{(y,y)})^2\\&=(\Vert x\Vert + \Vert y\Vert)^2\end{aligned} ∥ x + y ∥ 2 = ( x + y , x + y ) = ( x , x ) + ( x , y ) + ( y , x ) + ( y , y ) ≤ ( x , x ) + 2 ( x , x ) ( y , y ) + ( y , y ) = ( ( x , x ) + ( y , y ) ) 2 = ( ∥ x ∥ + ∥ y ∥ ) 2 ∥ x + y ∥ ≤ ∥ x ∥ + ∥ y ∥ \small \Vert x+y\Vert \leq \Vert x\Vert + \Vert y\Vert ∥ x + y ∥ ≤ ∥ x ∥ + ∥ y ∥ ∥ x ∥ = ( x , x ) \small \Vert x\Vert=\sqrt{(x,x)} ∥ x ∥ = ( x , x ) V \small V V ( V , ( ⋅ , ⋅ ) ) \small (V,\sqrt{(\,\cdot\,,\cdot\,)}) ( V , ( ⋅ , ⋅ ) )
R n R^n R n
到目前为止，已经介绍了线性空间、度量空间、赋范线性空间和内积空间，它们之间的关系如图所示
下面举个例子，R n = { ( ξ 1 , ξ 2 , ⋯ , ξ n ) ∣ ξ i ∈ R , i = 1 , 2 , ⋯ , n } \small R^n=\lbrace(\xi_1,\xi_2,\cdots,\xi_n)|\,\xi_i \in R,i=1,2,\cdots,n\rbrace R n = { ( ξ 1 , ξ 2 , ⋯ , ξ n ) ∣ ξ i ∈ R , i = 1 , 2 , ⋯ , n } R \small R R R n \small R^n R n ∀ x , y ∈ R n \small \forall \,x,y \in R^n ∀ x , y ∈ R n x = ( ξ 1 , ξ 2 , ⋯ , ξ n ) , y = ( η 1 , η 2 , ⋯ , η n ) x=(\xi_1,\xi_2,\cdots,\xi_n),\,y=(\eta_1,\eta_2,\cdots,\eta_n) x = ( ξ 1 , ξ 2 , ⋯ , ξ n ) , y = ( η 1 , η 2 , ⋯ , η n ) x x x y y y ( x , y ) ≜ ∑ i = 1 n ξ i η i = x T y = y T x (x,y)\triangleq\sum_{i=1}^n\xi_i\eta_i=x^Ty=y^Tx ( x , y ) ≜ i = 1 ∑ n ξ i η i = x T y = y T x ( R n , ( ⋅ , ⋅ ) ) \small (R^n,(\,\cdot\,,\cdot\,)) ( R n , ( ⋅ , ⋅ ) ) ∥ x ∥ = ( x , x ) = ∑ i = 1 n ξ i 2 = x T x \Vert x\Vert=\sqrt{(x,x)}=\sqrt{\sum_{i=1}^n\xi_i^2}=\sqrt{x^Tx} ∥ x ∥ = ( x , x ) = i = 1 ∑ n ξ i 2 = x T x 2 2 2 d ( x , y ) = ∥ x − y ∥ = ∑ i = 1 n ( ξ i − η i ) 2 d(x,y)=\Vert x-y\Vert=\sqrt{\sum_{i=1}^n{(\xi_i-\eta_i)}^2} d ( x , y ) = ∥ x − y ∥ = i = 1 ∑ n ( ξ i − η i ) 2 线性空间(Vector space) 度量空间(Metric space) 赋范线性空间(Normed vector space) 内积空间(Inner product space)