# Real Analysis RUPP 2019 (dragged)

```Rp
Rp
Rp
Rp = {x = (x1 , &middot; &middot; &middot; , xj , &middot; &middot; &middot; , xp ), x1 , &middot; &middot; &middot; , xj , &middot; &middot; &middot; , xp ∈ R}
λ&middot;x
x = (x1 , &middot; &middot; &middot; , xj , &middot; &middot; &middot; , xp )
λ
x
y
‫ذ‬+‫ذ‬
y = (y1 , &middot; &middot; &middot; , yj , &middot; &middot; &middot; , yp )
x+y
x
‫ذ&middot;ذ‬
Rp
λ
x + y = (x1 , &middot; &middot; &middot; , xj , &middot; &middot; &middot; , xp ) + (y1 , &middot; &middot; &middot; , yj , &middot; &middot; &middot; , yp )
= (x1 + y1 , &middot; &middot; &middot; , xj + yj &middot; &middot; &middot; , xp + yp )
λx = λ(x1 , &middot; &middot; &middot; , xj , &middot; &middot; &middot; , xp )
= (λx1 , &middot; &middot; &middot; , λxj , &middot; &middot; &middot; , λxp )
0 = (0, 0, &middot; &middot; &middot; , 0)
−1x = −x
RP
−(x1 &middot; &middot; &middot; , xp ) = −1(x1 &middot; &middot; &middot; , xp ) = (−x1 &middot; &middot; &middot; , −xp )
Rp
x, y
Rp
z
λ
&micro;
(x + y) + z = x + (y + z)
x+y =y+x
x+0=0+x=x
x + (−x) = (−x) + x = 0
λ(&micro;x) = (λ&micro;)x
(λ + &micro;)x = λx + &micro;x
λ(x + y) = λx + λy
1x = x
Rp
R
R
p
(R , +, &middot;)
p
R
p
R
x
λ
p
x = (x1 , &middot; &middot; &middot; , xj , &middot; &middot; &middot; , xp )
&micro;
y = (y1 , &middot; &middot; &middot; , yj , &middot; &middot; &middot; , yp )
λx + &micro;y = λ(x1 , &middot; &middot; &middot; , xj , &middot; &middot; &middot; , xp ) + &micro;(y1 , &middot; &middot; &middot; , yj , &middot; &middot; &middot; , yp )
= (λx1 + &micro;y1 , , &middot; &middot; &middot; , λxj + &micro;yj &middot; &middot; &middot; , λxp + &micro;yp )
Rp
Rp
S
x, y
Rp
z
S
(S, +, &middot;)
Rp
λ
S
&micro;
RP
S ⊂ Rp
Rp
(S, +, &middot;)
S &quot;= ∅
x, y ∈ S
x+y ∈S
λ∈R
x∈S
S ⊂ Rp
λx ∈ S
Rp
(S, +, &middot;)
S &quot;= ∅
λ, &micro; ∈ K
x, y ∈ S
a b
R
3
S = {(x, y, z) ∈ R3 , ax+by+cz = 0}
c
R3
S
λx + &micro;y ∈ S
S = {(x, y, z) ∈ R3 , z = 0}
a b
S
c
S = {(x, y, z) ∈ R3 ,
x y z
+ + = 1}
a b c
S
R3
y = (y1 , &middot; &middot; &middot; , yj , &middot; &middot; &middot; , yp )
%&middot;, &middot;&amp; : Rp &times;Rp → R
R
p
%x, y&amp; =
x, y, z
p
!
x = (x1 , &middot; &middot; &middot; , xj , &middot; &middot; &middot; , xp )
xj yj
j=1
Rp
λ, &micro;
%λx + &micro;y, z&amp; = λ%x, z&amp; + &micro;%y, z&amp;
R
RP
!x, λy + &micro;z&quot; = λ!x, y&quot; + &micro;!x, z&quot;
x
y
!x, y&quot; = !y, x&quot;
!x, x&quot; ≥ 0
Rp
x
x
Rp
Rp
!x, x&quot; = 0
Rp
!&middot;, &middot;&quot;
x, y
⇒
z
Rp
(Rp , !&middot;, &middot;&quot;)
!&middot;, &middot;&quot; : Rp &times; Rp → R
λ
x=0
Rp
Rp
&micro;
!λx + &micro;y, z&quot; = λ!x, z&quot; + &micro;!y, z&quot;
!x, λy + &micro;z&quot; = λ!x, y&quot; + &micro;!x, z&quot;
!x, y&quot; = !y, x&quot;
!x, x&quot; ≥ 0
!x, x&quot; = 0
⇒
(Rp , !&middot;, &middot;&quot;)
|!x, y&quot;| ≤
!
!x, x&quot;
x=0
x
!
!y, y&quot;
Rp
y
RP
x
y
c
x = cy
a1 , a 2 , &middot; &middot; &middot; , a n
|a1 b1 + a2 b2 + &middot; &middot; &middot; + an bn | ≤
λ
!
a21
+
y = cx
b1 , b 2 , &middot; &middot; &middot; , b n
a22
x
+ &middot;&middot;&middot; +
a2n
!
b21 + b22 + &middot; &middot; &middot; + b2n
y
&quot;λx + y, λx + y# ≥ 0
&quot;x, x#λ2 + &quot;x, y#λ + &quot;y, y# ≥ 0
∆! ≤ 0
∆! = (&quot;x, y#)2 − &quot;x, x#&quot;y, y#
(&quot;x, y#)2 ≤ &quot;x, x#&quot;y, y#
|&quot;x, y#| ≤
x
Rp
&quot;
&quot;
&quot;x, x# &quot;y, y#
(Rp , &quot;&middot;, &middot;#)
||x|| :=
|| &middot; || : Rp → R
&quot;
&quot;x, x#
Rp
x, y
|| &middot; ||
(Rp , || &middot; ||)
λ
||x|| = 0
||λx|| = |λ|||x||
||x + x|| ≤ ||x|| + ||y||
Rp
&quot;&middot;, &middot;#
Rp
x ∈ Rp
||x|| = 0 ⇒
⇒
⇒
x ∈ Rp
λ∈R
RP
!
&quot;x, x# = 0
&quot;x, x# = 0
x=0
!
||λx|| =
&quot;λx, λx#
!
=
λ&quot;x, λx#
!
=
λλ&quot;x, x#
!
= |λ| &quot;x, x#
=
|λ|||x||
(Rp , &quot;&middot;, &middot;#)
x
y
Rp
&amp;x + y&amp;2 = &amp;x&amp;2 + &amp;y&amp;2 + 2&quot;x, y#
|| &middot; ||
&quot;&middot;, &middot;#
&amp;x − y&amp;2 = &amp;x&amp;2 + &amp;y&amp;2 − 2&quot;x, y#
RP
!
&quot;
2 !x!2 + !y!2 = !x + y!2 + !x − y!2
&quot;
1!
!x + y!2 − !x!2 − !y!2
2
&quot;
1!
#x, y\$ =
!x + y!2 − !x − y!2
4
#x, y\$ =
(Rp , #&middot;, &middot;\$)
x
y
Rp
#&middot;, &middot;\$
θ
|| &middot; ||
#x, y\$ = ||x||||y|| cos θ
θ
x
y
(Rp , #&middot;, &middot;\$)
x
y
Rp
#x, y\$ = 0
x⊥y
A
Rp
x⊥A
Rp
x
y
A
#x, y\$ = 0
A
B
A⊥B
Rp
(x, x)
A
B
A&times;B
#x, y\$ = 0
X
Rp
A
X Rp
X ⊥ = {y ∈ Rp , #x, y\$ = 0, x ∈ X}
X⊥
Rp
B = (v1 , v2 , &middot; &middot; &middot; , vj , &middot; &middot; &middot; , vp )
!vi , vj &quot; = 0,
RP
i #= j, (i, j) ∈ [1, p] &times; [1, p]
Rp
B = (v1 , v2 , &middot; &middot; &middot; , vj , &middot; &middot; &middot; , vp )
!vi , vj &quot; = δij ,
(i, j) ∈ [1, p] &times; [1, p]
δij

 1
δij =
 0
i=j,
i #= j
B = (v1 , v2 , &middot; &middot; &middot; , vj , &middot; &middot; &middot; , vp )
Rp
x = (x1 , &middot; &middot; &middot; , xi , &middot; &middot; &middot; , xj , &middot; &middot; &middot; , xp )
y = (y1 , &middot; &middot; &middot; , yi , &middot; &middot; &middot; , yj , &middot; &middot; &middot; , yp )
!x, y&quot; =
p
p
\$
\$
i=1 j=1
!vi , vj &quot;xi yj
x
y
Rp
%
&amp;
||x + y||2 + ||x − y||2 = 2 ||x||2 + ||y||2
x
x⊥y
⇔
y
Rp
Rp
||x + y||2 = ||x||2 + ||y||2
R2
Φ
x = (x1 , x2 )
Φ(x, y) = 2x1 y1 + 2x2 y2 + x1 y2 + x2 y1
y = (y1 , y2 )
RP
R2
Ψ
x = (x1 , x2 , x3 )
y = (y1 , y2 , y3 )
R3
Φ(x, y) = 2x1 y1 + 2x2 y2 + 2x3 y3 + x1 y2 + x2 y1 + x1 y3 + x3 y1 + x2 y3 + x3 y2
R3
a
Φ
x = (x1 , y1 )
y = (y1 , y2 )
R2
Φ(x, y) = x1 y1 + x2 y2 + a(x1 y2 + x2 y1 )
a
R2
Φ
x = (x1 , x2 , &middot; &middot; &middot; , xp )
y = (y1 , y2 , &middot; &middot; &middot; , yp )
Rp
ϕ(x, y) = w1 x1 y1 + w2 x2 y2 + &middot; &middot; &middot; + wp xp yp
wj &gt; 0
ϕ
j = 1, 2, &middot; &middot; &middot; , p
(Rp , !&middot;, &middot;&quot;)
x
y
Rp
R
p
|| &middot; ||
!
!
! x
!
y
\$x − y\$
!
!
−
! ||x||2 ||y||2 ! = ||x|| &middot; ||y||
!&middot;, &middot;&quot;
RP
(Rp , +, &middot;)
R
x
y
Rp
Rp
Rp
⇔
x=0
||x + y|| ≤ ||x|| + ||y||
|| &middot; || : Rp → R
x
Rp
||x|| ≥ 0
x
||λx|| = |λ|||x||
Rp
|| &middot; ||
R
λ
||x|| = 0
(Rp , || &middot; ||)
Rp
|| &middot; ||
Rp
||x|| &lt; 0
N
'
'
'
'
T&copy;QBSBUJPO BOE &amp; TFQBSBUJPO
IPNPH&copy;O&copy;JU&copy; BOE &amp; IPNPHFOFJUZ
TPVTBEEJUJWJU&copy; BOE &amp; TVC BEEJUJWJUZ
JO&copy;HBMJU&copy; USJBOHVMBJSF BOE &amp; USJBOHMF JOFRVBMJUZ
OPSN
RP
Rp
|| &middot; ||
x ∈ Rp
0 = x + (−x)
||0|| =
≤
≤
≤
||0|| = 0
y
||x + (−x)||
||x|| + || − x||
||x|| + | − 1|||x||
2||x||
||x|| ≥ 0
Rp
(Rp , || &middot; ||)
!
!
!
!
! ||x|| − ||y||! ≤ ||x − y||
x
||x|| = ||(x − y) + y||
≤ ||x − y|| + ||y||
||x|| − ||y|| ≤ ||x − y||
x
||y|| − ||x|| ≤ ||y − x||
y
||y − x|| = || − (x − y)|| = | − 1|||x − y||
− (||x|| − ||y||) ≤ ||x − y||
−||x − y|| ≤ ||x|| − ||y||
−||x − y|| ≤ ||x|| − ||y|| ≤ ||x − y||
RP
(Rp , || &middot; ||)
x
||x|| = 1
x
(Rp , || &middot; ||)
u=
1
x
||x||
Rp x = (x1 , &middot; &middot; &middot; , xj , &middot; &middot; &middot; , xp )
||x||1 =
||x||2 =
u
&quot;
p
!
k=1
p
!
k=1
|xk |
|xk |2
# 12
||x||∞ = max |xk |
1≤k≤p
|| &middot; ||1 || &middot; ||2
|| &middot; ||∞
x = (x1 , &middot; &middot; &middot; , xj , &middot; &middot; &middot; , xp )
R
J
Rp
Rp
y = (y1 , &middot; &middot; &middot; , yj , &middot; &middot; &middot; , yp )
||x||1 = |x1 | + &middot; &middot; &middot; + |xp |
||x||1 = 0
|x1 | + &middot; &middot; &middot; + |xp | = 0
x1 = &middot; &middot; &middot; = xp = 0
x=0
||0|| = 0
Rp
λ
RP
JJ
||λx||1 = |λx1 | + &middot; &middot; &middot; + |λxp |
= |λ| (|x1 | + &middot; &middot; &middot; + |xp |)
= |λ|||x||1
JJJ
||x + y||1 =
≤
≤
≤
J
||x||2 =
|x1 + y1 | + &middot; &middot; &middot; + |xp + yp |
(|x1 | + |y1 |) + &middot; &middot; &middot; + (|xp | + |yp |)
(|x1 | + &middot; &middot; &middot; + |xp |) + (|y1 | + &middot; &middot; &middot; + |yp |)
||x||1 + ||y||1
!
x21 + x22 + &middot; &middot; &middot; + x2p
||x||2 = 0
||x||22 = 0
x21 + x22 + &middot; &middot; &middot; + x2p = 0
x1 = x2 = &middot; &middot; &middot; = xp = 0
x=0
x=0
||x||2 = 0
JJ
||λx||2
!
=
(λx )2 + &middot; &middot; &middot; + (λxn )2
! &quot;1
#
=
λ2 x21 + &middot; &middot; &middot; + x2p
!
= |λ| x21 + &middot; &middot; &middot; + x2p
= |λ|||x||2
JJJ
||x + y||22 =
≤
≤
≤
≤
|x1 + y1 |2 + &middot; &middot; &middot; + |xp + yp )2
(|x1 | + |y1 |)2 + &middot; &middot; &middot; + (|xp | + |yp |)2
(|x1 |2 + 2|x1 ||y1 | + |y1 |2 ) + &middot; &middot; &middot; + (|xp |2 + 2|xp ||yp | + |y 2 |p )
(|x1 |2 + &middot; &middot; &middot; + |xp |2 ) + 2 (|x1 ||y1 | + &middot; &middot; &middot; + |xp ||yp |) + (|y1 |2 + &middot; &middot; &middot; + |yp |2 )
||x||22 + 2 (|x1 ||y1 | + &middot; &middot; &middot; + |xp ||yp |) + &middot; &middot; &middot; + ||y||22
|x1 ||y1 | + &middot; &middot; &middot; + |xp ||yp | = |x1 ||y1 | + &middot; &middot; &middot; + |xp ||yp |
RP
&quot;
!
2
2
|x1 ||y1 | + &middot; &middot; &middot; + |xp ||yp | ≤
|x1 | + &middot; &middot; &middot; + |xp | |y1 |2 + &middot; &middot; &middot; + |y|2p
≤ ||x||2 ||y||2
||x + y||22 ≤ ||x||22 + 2||x||2 ||y||2 + ||y||22
≤ (||x||2 + ||y||2 )2
||x + y||2 ≤ ||x||2 + ||y||2
J
JJ
||x||∞ = max {|x1 |, &middot; &middot; &middot; , |xp |}
||x||∞ = 0
|x1 | = &middot; &middot; &middot; = |xp | = 0
x1 = &middot; &middot; &middot; = xp = 0
x=0
x=0
||x||∞ = 0
||λx||∞ =
=
=
=
max {|λx1 |, &middot; &middot; &middot; , |λxp |}
max {|λ||x1 |, &middot; &middot; &middot; , |λ||xp |}
|λ| max {|x1 |, &middot; &middot; &middot; , |xp |}
|λ|||x||∞
JJJ
||x + y||∞ =
≤
≤
≤
Rp
|| &middot; ||1
max {|x1 + y1 | , &middot; &middot; &middot; , |xp + yp |}
max {|x1 | + |y1 |, &middot; &middot; &middot; , |xp | + |yp |}
max {|x1 |, &middot; &middot; &middot; , |xp |} + max {|y1 |, &middot; &middot; &middot; , |yp |}
||x||∞ + ||y||∞
|| &middot; ||1
|| &middot; ||2
|| &middot; ||2
|| &middot; ||1 ≤ || &middot; ||2
Rp
C
RP
Rp
x
||x||1 ≤ C||x||2
R
p
|| &middot; ||2
|| &middot; ||1
|| &middot; ||2
|| &middot; ||1
|| &middot; ||1 ∼ || &middot; ||2 || &middot; ||1
C1
C2
C1 ||x||2 ≤ ||x||1 ≤ C2 ||x||2
R
|| &middot; ||1 , || &middot; ||2
p
Rp
x
||x||∞ ≤ ||x||1 ≤ p||x||∞
√
||x||∞ ≤ ||x||2 ≤ p||x||∞
1
√
√ ||x||2 ≤ ||x||1 ≤ p||x||2
p
Rp
x = (x1 , &middot; &middot; &middot; , xp )
|| &middot; ||∞
|| &middot; ||1
|| &middot; ||∞
||x||∞ =
≤
max |xk |
1≤k≤p
p
!
k=1
|xk |
||x||∞ ≤ ||x||1
||x||1 =
≤
|| &middot; ||2
x
p
!
k=1
|xk |
max |xk | + max |xk | + &middot; &middot; &middot; + max |xk |
1≤k≤p
1≤k≤p
1≤k≤p
||x||1 ≤ p||x||∞
||x||∞ ≤ ||x||1 ≤ p||x||∞
|| &middot; ||2
|| &middot; ||∞
||x||2∞
=
=
≤
!
max |xk |
1≤k≤p
max |xk |2
&quot;2
1≤k≤p
p
#
k=1
|xk |2
≤ ||x||22
||x||∞ ≤ ||x||2
||x||22
=
p
#
k=1
≤
|xk |2
max |xk |2 + max |xk |2 + &middot; &middot; &middot; + max |xk |2
1≤k≤p
1≤k≤p
1≤k≤p
≤ p max |xk |2
≤
||x||2 ≤
√
1≤k≤p
p||x||2∞
p||x||∞
||x||∞ ≤ ||x||2 ≤
|| &middot; ||1
√
p||x||∞
|| &middot; ||2
||x||21 ≤ p||x||22
||x||1 ≤
√
p||x||2
RP
RP
1
√ ||x||2 ≤ ||x||1
p
1
√
√ ||x||2 ≤ ||x||1 ≤ p||x||2
p
|| &middot; ||
p = 1
x
||x||1 = ||x||2 = ||x||∞ = |x|
Rp
(Rp , ||&middot;||)
d : Rp &times;Rp → R
d(x, y) = ||x − y||
x, y
z
Rp
d(x, y) = 0
⇔
x=y
d(x, y) = d(y, x)
d(x, z) ≤ d(x, y) + d(y, z)
Rp
d
x, y
z
|| &middot; ||
RP
(Rp , d)
Rp
Rp
d(x, y) =
=
=
=
=
||x − y||
|| − (y − x)||
| − 1|||y − x||
||y − x||
d(y, x)
d(x, y) = 0 ⇔ ||x − y|| = 0
⇔ x−y =0
⇔ x=y
d(x, z) =
=
≤
≤
Rp
x, y
z
Rp
||x − z||
||(x − y) + (y − z)||
||x − y|| + ||y − z||
δ(x, y) + δ(y, z)
d : Rp &times; R p → R
d(x, y) = 0
⇔
x=y
Rp
d(x, y) = d(y, x)
d(x, z) ≤ d(x, y) + d(y, z)
RP
Rp
(Rp , d)
x
Rp
y
d(x, y) =
Rp
d

0
x=y
1
x != y
(Rp , d)
x
y
d(x, y) ≥ 0
x
y
Rp
Rp
d : Rp &times; Rp → R
d(x, y) &lt; 0
Rp
d
y =z
0 ≤ 2d(x, y)
(Rp , d)
d(x, y)
x
y
d(x, x) ≤ (x, y) + d(y, x)
d(x, y) ≥ 0
x
y
```