Norme na
R
2
{\displaystyle R^{2}}
funkcije na
R
2
{\displaystyle R^{2}}
p-norme možemo definisati za svaki realan broj
p
⩾
1
,
{\displaystyle p\geqslant 1,}
p
=
∞
{\displaystyle p=\infty }
2-norma standardna euklidska norma
1-norma je poznata pod nazivom taxicab- norma a
∞
{\displaystyle \infty }
max-norma .
Svaka norma definiše udaljenost (metriku). Uvođenjem p-normi dobili smo razne načine za računanje udaljenosti između dvije tačke.
Definicija 1
uredi
Realna funkcija
‖
∗
‖
{\displaystyle \rVert \ast \lVert }
R
2
→
R
{\displaystyle R^{2}\rightarrow R}
2 ako ima sljedeće osobine:
‖
(
x
,
y
)
‖
⩾
0
{\displaystyle \rVert (x,y)\lVert \geqslant 0}
‖
(
x
,
y
)
‖
=
0
<=>
(
x
,
y
)
=
0
{\displaystyle \rVert (x,y)\lVert =0<=>(x,y)=0}
‖
(
x
,
y
)
+
(
u
,
v
)
‖
≤
0
<=>
‖
(
x
,
y
)
‖
+
‖
+
(
u
,
v
)
‖
{\displaystyle \rVert (x,y)+(u,v)\lVert \leq 0<=>\rVert (x,y)\lVert +\rVert +(u,v)\lVert }
‖
c
(
x
,
y
)
‖
=
|
c
|
‖
(
x
,
y
)
‖
{\displaystyle \rVert c(x,y)\lVert =|c|\rVert (x,y)\lVert }
Za sve
(
x
,
y
)
,
(
u
,
v
)
∈
R
2
{\displaystyle (x,y),(u,v)\in R^{2}}
c
∈
R
{\displaystyle c\in R}
R
2
{\displaystyle R^{2}}
R
2
{\displaystyle R^{2}}
Primjer norme na
R
2
{\displaystyle R^{2}}
dužinu dužine čije su krajnje tačke (
0
,
0
)
{\displaystyle 0,0)}
(
x
,
y
)
{\displaystyle (x,y)}
‖
(
x
,
y
)
‖
p
=
x
2
+
y
2
{\displaystyle \rVert (x,y)\lVert _{p}={\sqrt {x^{2}+y^{2}}}}
koja predstavlja poseban slučaj p-normi
‖
(
x
,
y
)
‖
p
{\displaystyle \rVert (x,y)\lVert _{p}}
Definicija 2
uredi
Za svaki realni broj p,
p
⩾
1
{\displaystyle p\geqslant 1}
(
‖
(
x
,
y
)
‖
p
=
|
x
|
p
+
|
y
|
p
)
1
p
{\displaystyle (\rVert (x,y)\lVert _{p}=|x|^{p}+|y|^{p})^{\frac {1}{p}}}
Funkcija
‖
(
x
,
y
)
‖
p
{\displaystyle \rVert (x,y)\lVert _{p}}
R
2
{\displaystyle R^{2}}
definicije 1.
Osobine (1), (2) i (4) se lako provjere, dok je provjera (3) poznata kao nejednakost trougla , za ovaj dokaz koristi se Youngova nejednakost.
Za sve
a
,
b
⩾
0
i
p
,
q
>
1
{\displaystyle a,b\geqslant 0ip,q>1}
1
p
+
1
q
=
1
{\displaystyle {\frac {1}{p}}+{\frac {1}{q}}=1}
a
b
⩽
a
p
p
+
b
q
q
{\displaystyle ab\leqslant {\frac {a^{p}}{p}}+{\frac {b^{q}}{q}}}
Za svaki
p
⩾
1
{\displaystyle p\geqslant 1}
‖
(
x
,
y
)
+
(
u
+
v
)
‖
p
≤
‖
(
x
,
y
)
‖
p
+
‖
(
u
,
v
)
‖
p
p
{\displaystyle \rVert (x,y)+(u+v)\lVert _{p}\leq {\sqrt[{p}]{\rVert (x,y)\lVert _{p}+\rVert (u,v)\lVert _{p}}}}
Dokaz
Treba dokazati da za svaki
p
⩾
1
{\displaystyle p\geqslant 1}
‖
(
x
,
y
)
+
(
u
+
v
)
‖
p
≤
‖
(
x
,
y
)
‖
p
+
‖
(
u
,
v
)
‖
p
{\displaystyle \rVert (x,y)+(u+v)\lVert _{p}\leq \rVert (x,y)\lVert _{p}+\rVert (u,v)\lVert _{p}}
tj.
|
x
+
u
|
p
+
|
y
+
v
|
p
p
≤
|
x
|
p
+
|
u
|
p
p
+
|
y
|
p
+
|
u
v
|
p
p
{\displaystyle {\sqrt[{p}]{|x+u|^{p}+|y+v|^{p}}}\leq {\sqrt[{p}]{|x|^{p}+|u|^{p}}}+{\sqrt[{p}]{|y|^{p}+|uv|^{p}}}}
za sve
(
x
,
y
)
,
(
u
,
v
)
∈
R
2
{\displaystyle (x,y),(u,v)\in R^{2}}
Za
p
=
1
{\displaystyle p=1}
[
|
x
+
u
|
≤
|
x
|
+
|
u
|
∧
[
|
y
+
v
|
≤
|
y
|
+
|
v
|
]
=>
|
x
+
u
|
+
|
y
+
v
|
≤
|
x
|
+
|
u
|
+
|
y
|
+
|
v
|
{\displaystyle [|x+u|\leq |x|+|u|\land [|y+v|\leq |y|+|v|]=>|x+u|+|y+v|\leq |x|+|u|+|y|+|v|}
Za
p
>
1
{\displaystyle p>1}
U nejednakosti
a
b
⩽
a
p
p
+
b
q
q
{\displaystyle ab\leqslant {\frac {a^{p}}{p}}+{\frac {b^{q}}{q}}}
uvrstit ćemo izraze za
a
{\displaystyle a}
b
{\displaystyle b}
a
=
|
x
|
‖
(
x
,
y
)
‖
p
{\displaystyle a={\frac {|x|}{\rVert (x,y)\lVert _{p}}}}
b
=
|
u
|
‖
(
u
,
v
)
‖
q
{\displaystyle b={\frac {|u|}{\rVert (u,v)\lVert _{q}}}}
a
=
|
y
|
‖
(
x
,
y
)
‖
p
{\displaystyle a={\frac {|y|}{\rVert (x,y)\lVert _{p}}}}
b
=
|
v
|
‖
(
u
,
v
)
‖
q
{\displaystyle b={\frac {|v|}{\rVert (u,v)\lVert _{q}}}}
dobijamo nejednakosti
|
x
|
‖
(
x
,
y
)
‖
p
∗
|
u
|
‖
(
u
,
v
)
‖
q
⩽
|
x
|
p
p
‖
(
x
,
y
)
‖
p
∗
|
u
|
q
q
‖
(
u
,
v
)
‖
q
{\displaystyle {\frac {|x|}{\rVert (x,y)\lVert _{p}}}*{\frac {|u|}{\rVert (u,v)\lVert _{q}}}\leqslant {\frac {|x|^{p}}{p\rVert (x,y)\lVert _{p}}}*{\frac {|u|^{q}}{q\rVert (u,v)\lVert _{q}}}}
|
y
|
‖
(
x
,
y
)
‖
p
∗
|
v
|
‖
(
u
,
v
)
‖
q
⩽
|
y
|
p
p
‖
(
x
,
y
)
‖
p
∗
|
v
|
q
q
‖
(
u
,
v
)
‖
q
{\displaystyle {\frac {|y|}{\rVert (x,y)\lVert _{p}}}*{\frac {|v|}{\rVert (u,v)\lVert _{q}}}\leqslant {\frac {|y|^{p}}{p\rVert (x,y)\lVert _{p}}}*{\frac {|v|^{q}}{q\rVert (u,v)\lVert _{q}}}}
saberemo li ih dobijamo nejednakosti
|
x
|
|
u
|
+
|
y
|
|
v
|
‖
(
x
,
y
)
‖
p
‖
(
u
,
v
)
‖
q
⩽
|
x
|
p
+
|
y
|
p
p
‖
(
x
,
y
)
‖
p
+
|
u
|
q
+
|
v
|
q
q
‖
(
u
,
v
)
‖
q
=
1
p
+
1
q
=
1
{\displaystyle {\frac {|x||u|+|y||v|}{\rVert (x,y)\lVert _{p}\rVert (u,v)\lVert _{q}}}\leqslant {\frac {|x|^{p}+|y|^{p}}{p\rVert (x,y)\lVert _{p}}}+{\frac {|u|^{q}+|v|^{q}}{q\rVert (u,v)\lVert _{q}}}={\frac {1}{p}}+{\frac {1}{q}}=1}
tj
|
x
|
|
u
|
+
|
y
|
|
v
|
⩽
x
p
+
y
p
q
+
u
q
+
v
q
p
{\displaystyle |x||u|+|y||v|\leqslant {\sqrt[{q}]{x^{p}+y^{p}}}+{\sqrt[{p}]{u^{q}+v^{q}}}}
x
,
y
,
u
,
v
∈
R
{\displaystyle x,y,u,v\in R}
(
x
+
u
)
p
−
1
{\displaystyle (x+u)^{p-1}}
u
{\displaystyle u}
y
+
v
)
q
−
1
{\displaystyle y+v)^{q-1}}
v
{\displaystyle v}
(
p
−
1
)
q
=
p
{\displaystyle (p-1)_{q}=p}
|
x
|
|
x
+
u
|
p
−
1
+
|
y
|
|
y
+
v
|
p
−
1
⩽
x
p
+
y
p
p
+
u
q
+
v
q
q
{\displaystyle |x||x+u|^{p-1}+|y||y+v|^{p-1}\leqslant {\sqrt[{p}]{x^{p}+y^{p}}}+{\sqrt[{q}]{u^{q}+v^{q}}}}
Zamjenom uloga x i u , y i v imamo
|
u
|
|
x
+
u
|
p
−
1
+
|
v
|
|
y
+
v
|
p
−
1
⩽
u
p
+
v
p
p
+
|
x
+
u
|
p
+
|
y
+
p
|
p
q
{\displaystyle |u||x+u|^{p-1}+|v||y+v|^{p-1}\leqslant {\sqrt[{p}]{u^{p}+v^{p}}}+{\sqrt[{q}]{|x+u|^{p}+|y+p|^{p}}}}
|
x
+
u
|
p
+
|
y
+
v
|
p
=
|
x
+
u
|
|
x
+
u
|
p
−
1
+
|
y
+
v
|
|
y
+
v
|
p
−
1
{\displaystyle |x+u|^{p}+|y+v|^{p}=|x+u||x+u|^{p-1}+|y+v||y+v|^{p-1}}
⩽
(
|
x
|
+
|
u
|
|
x
+
u
|
p
−
1
+
(
|
y
|
+
|
v
|
)
|
y
+
v
|
p
−
1
=
{\displaystyle \leqslant (|x|+|u||x+u|^{p-1}+(|y|+|v|)|y+v|^{p-1}=}
|
x
|
|
x
+
u
|
p
−
1
+
|
u
|
|
x
+
u
|
p
−
1
+
|
y
|
|
y
+
v
|
p
−
1
+
|
v
|
|
y
+
v
|
p
−
1
=
{\displaystyle |x||x+u|^{p-1}+|u||x+u|^{p-1}+|y||y+v|^{p-1}+|v||y+v|^{p-1}=}
na osnovu ranijih nejednakosti imamo
|
x
|
p
+
|
y
|
p
p
|
x
+
u
|
p
+
|
y
+
v
|
p
q
+
|
u
|
p
+
|
v
|
p
p
|
x
+
u
|
p
+
|
y
+
v
|
p
q
{\displaystyle {\sqrt[{p}]{|x|^{p}+|y|^{p}}}{\sqrt[{q}]{|x+u|^{p}+|y+v|^{p}}}+{\sqrt[{p}]{|u|^{p}+|v|^{p}}}{\sqrt[{q}]{|x+u|^{p}+|y+v|^{p}}}}
odnosno
|
x
+
u
|
p
+
|
y
+
v
|
p
⩽
(
|
x
|
p
+
|
y
|
p
p
+
|
u
|
p
+
|
v
|
p
p
)
(
|
x
+
u
|
p
+
|
y
+
v
|
p
)
)
{\displaystyle |x+u|^{p}+|y+v|^{p}\leqslant ({\sqrt[{p}]{|x|^{p}+|y|^{p}}}+{\sqrt[{p}]{|u|^{p}+|v|^{p}}})(|x+u|^{p}+|y+v|^{p}))}
Dijeljenjem s drugim faktorom s desne strane slijedi
|
x
+
u
|
p
+
|
y
+
v
|
p
n
⩽
(
|
x
|
p
+
|
y
|
p
p
+
|
u
|
p
+
|
v
|
p
p
)
{\displaystyle {\sqrt[{n}]{|x+u|^{p}+|y+v|^{p}}}\leqslant ({\sqrt[{p}]{|x|^{p}+|y|^{p}}}+{\sqrt[{p}]{|u|^{p}+|v|^{p}}})}
![{\displaystyle \rVert (x,y)+(u+v)\lVert _{p}\leq {\sqrt[{p}]{\rVert (x,y)\lVert _{p}+\rVert (u,v)\lVert _{p}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/240c1a04d0dabf84bbc24ac72ea1fd79532a26ce)
![{\displaystyle {\sqrt[{p}]{|x+u|^{p}+|y+v|^{p}}}\leq {\sqrt[{p}]{|x|^{p}+|u|^{p}}}+{\sqrt[{p}]{|y|^{p}+|uv|^{p}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bce191a8b908759b280fd7870e648cda0005bc4a)
![{\displaystyle [|x+u|\leq |x|+|u|\land [|y+v|\leq |y|+|v|]=>|x+u|+|y+v|\leq |x|+|u|+|y|+|v|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bca9d9d6503718b75388a76cde46000cc5317eaf)
![{\displaystyle |x||u|+|y||v|\leqslant {\sqrt[{q}]{x^{p}+y^{p}}}+{\sqrt[{p}]{u^{q}+v^{q}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ca787f472d076e1f068b3601418f5b44406fd833)
![{\displaystyle |x||x+u|^{p-1}+|y||y+v|^{p-1}\leqslant {\sqrt[{p}]{x^{p}+y^{p}}}+{\sqrt[{q}]{u^{q}+v^{q}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0e2c85837b8939a63aa22a3b205f311c0d3e0be9)
![{\displaystyle |u||x+u|^{p-1}+|v||y+v|^{p-1}\leqslant {\sqrt[{p}]{u^{p}+v^{p}}}+{\sqrt[{q}]{|x+u|^{p}+|y+p|^{p}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3941bc75e053601c9e6861187a798df767f29f8e)
![{\displaystyle {\sqrt[{p}]{|x|^{p}+|y|^{p}}}{\sqrt[{q}]{|x+u|^{p}+|y+v|^{p}}}+{\sqrt[{p}]{|u|^{p}+|v|^{p}}}{\sqrt[{q}]{|x+u|^{p}+|y+v|^{p}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/af2e08404a344775ea6ecfb67d3823f22c3d2941)
![{\displaystyle |x+u|^{p}+|y+v|^{p}\leqslant ({\sqrt[{p}]{|x|^{p}+|y|^{p}}}+{\sqrt[{p}]{|u|^{p}+|v|^{p}}})(|x+u|^{p}+|y+v|^{p}))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/982ceacd127caaee461989a8a272fdc6ac396dc0)
![{\displaystyle {\sqrt[{n}]{|x+u|^{p}+|y+v|^{p}}}\leqslant ({\sqrt[{p}]{|x|^{p}+|y|^{p}}}+{\sqrt[{p}]{|u|^{p}+|v|^{p}}})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/60bf00d9158f460359c2299f2566b3aa1d8f06f6)
