SUMMARY

Question 1

YOUNGS INEQUALITY

Answer 1

let 1/p + 1/q = 1 with 1<p< ∞.
if a,b>=0,
then ab<= (a^p)/p + (b^q)/q

Question 2

HOLDERS INEQUALITY

Answer 2

let x,y∈K^n and 1/p + 1/q = 1 with 1<p< ∞
Σ|xiyi|<= (Σ|xi|^p)^1/p (Σ|yi|^q)^1/q
*sums from i=1 to n

Question 3

MINKOWSKIS INEQUALITY

Answer 3

let x,y∈K^n and 1/p + 1/q = 1 with 1<=p< ∞
(Σ|xi+yi|^p)^1/p <= (Σ|xi|^p)^1/p + (Σ|yi|^p)^1/p
*sums from i=1 to n

Question 4

DISTANCE BETWEEN X∈X AND V
(V subset of NLS X)

Answer 4

d(x,V) := inf{||x-v||: v ∈ V}

Question 5

CLOSURE of V
(V subset of NLS X)

Answer 5

V(-) := {x∈X: d(x,V)=0}

Question 6

V IS OPEN
(V subset of NLS X)

Answer 6

∀x∈V, ∃ε>0: B(x;ε) := {y∈X :||x-y||< ε} ⊆V

Question 7

DENSE

Answer 7

the subset E ⊆X when E(-)=X
where X is a metric space

Question 8

SEPARABLE

Answer 8

metric space X if it contains a countable dense subset

Question 9

CAUCHY-SCHWARZ INEQUALITY

Answer 9

if X is an IPS then:
1. |<x,y>|^2 <= <x,x><y,y>
2. ||x|| = √<x,x> is a norm
3. xn -> x, yn -> y ⟹ <xn,yn> -> <x,y>

Question 10

PARALLELOGRAM LAW

Answer 10

||x+y||^2 + ||x-y||^2 = 2(||x||^2 + ||y||^2)

Question 11

POLARISATION IDENTITY (K=R)

Answer 11

4<x,y> = ||x+y||^2 - ||x-y||^2

Question 12

POLARISATION IDENTITY (K=C)

Answer 12

4<x,y> = ||x+y||^2 - ||x-y||^2 + i||x+iy||^2 - i||x-iy||^2

Question 13

x⊥y

Answer 13

if <x,y>=0

Question 14

PYTHAGOREAM THEOREM

Answer 14

x⊥y ⟹ ||x+y||^2 = ||x||^2 + ||y||^2

Question 15

ORTHOGONAL COMPLEMENT OF V

Answer 15

V^⊥ = {x∈X: <x,v>=0 for all v∈V} is a closed linear subspace
where V subset of IPS X

Question 16

BESSEL’S INEQUALITY

Answer 16

assume X is IPS, {ei:i ∈N} is orthonormal set
then for all x ∈X we have
Σ|<x,ei>|^2 <= ||x||^2

*series on left converges

Question 17

CONVERGENCE OF SERIES THEOREM

Answer 17

if X hilbert with ONS {ei:i ∈N} then
Σλiei converges in X ⟺ Σ|λi|^2<∞
if either holds then
||Σλiei||^2 = Σ|λi|^2

Question 18

EXISTENCE THEOREM

Answer 18

if X is ∞-dimensional hilbert space then X has orthonormal basis
⟺ X is seperable

Question 19

CLOSEDNESS THEOREM

Answer 19

X NLS and Y banach ⟹ K(X,Y) closed in B(X,Y)

Question 20

||e^ix|| =

Answer 20

1