Norms, Metrics and Topologies

conditions to be a norm

• ‖x‖ =0 iff x=0

• ‖λx‖ = λ‖x‖

• ‖x+y‖ ≤ ‖x‖ +‖y‖ (triangle inequality)

how to define infinity norm ie. ‖x‖_∞=…

‖x‖_∞= max (x₁, x₂, …, x_n)

how to define p-norm ie ‖x‖_p=…

‖x‖_p= ( ∑ |x_i|^p ) ^1/p

what is a normed space

a pair (X, ‖^.‖) where X is a vector space and ‖^.‖ is a norm

when is a set K convex

∀ x,y∈K, λx + (1-λ)y ∈K

what is Minkowksi’s inequality

‖x+y‖_p ≤ ‖x‖_p + ‖y‖_p

when are two norms ‖·‖ and ‖·‖’ on X equivalent

if ∃ 0<c₁ ≤ c₂ s.t
c₁ ‖x‖ ≤ ‖x‖’ ≤ c₂ ‖x‖ ∀x∈X

what is the space ℓ^p , what is its (usual) norm and what kind of space is it

• space of all the sequences (x_j) st ∑ |x_i|^p converges

• equipped with the p-norm

• infinite dimensional vector space

what is the space ℓ^∞ and what is its (usual) norm

• space of bounded sequences

• equipped with the infinity norm

what is the space C([a,b]) and its (usual) norm

• the space of continuous functions on [a,b]

• sup-norm ie ‖f‖_∞= sup_x∈[a,b] |f(x)|

if T: X→Y, what is the operator norm on T

‖T‖= sup { (‖T(x)‖_Y) / (‖x‖_X) | x∈X, x≠0 }

conditions for a metric d on a set X

• d(x,y) = 0 iff x=y

• d(x,y) = d(y,x)

• d(x,y) ≤ d(x,z) + d(z,y) (triangle inequality)

what is the discrete metric

d(x,y)= 0 if x=y, =1 otherwise

let (X₁,d₁) and (X₂,d₂) be metric spaces. For any 1 ≤ p < ∞, how can we define a metric on X₁×X₂, (then also for p=∞)

• ϱ_p( (x₁, x₂),(y₁,y₂) ) = ( d₁(x₁,y₁)^p + d₂(x₂,y₂)^p )^1/p

• ϱ_∞( (x₁, x₂),(y₁,y₂) ) = max( d₁(x₁,y₁),d₂(x₂,y₂) )

when is S⊂(X,d) bounded

when ∃ a∈X and r>0 s.t S⊂B(a,r)

when is U⊂(X,d) open (in X). when is it closed

• when ∀ x∈U, ∃ ε>0 st B(x,ε)⊂U

• closed when X\U is open

what is true about an intersections/unions of finitely many open/closed sets (does it hold for countably many?)

• intersections, open: open, not for infinite

• intersections, closed: closed, yes for infinite

• unions, open: open, yes for infinite

• unions, closed: closed, not for infinite

how can we define convergence of (x_n)∈(X,d) in terms of open sets

x_n→x iff for every open set U that contains x, ∃ N≥1 s.t x_n∈U for all n≥N

how can we define closedness using convergent sequences

F⊆(X,d) is closed iff for every convergent (x_n)∈F, the limit is also found in F

when is a function f: X→Y Lipschitz cts

if ∃ C≥0 s.t
d_Y( f(x), f(y) ) ≤ C d_X(x,y) for ∀ x,y∈X

give an example of a Lipschitz function

d(x,A)= inf_z∈A d(x,z)

let X and Y be metric spaces. when is f:X→Y cts (using sets)

f is cts iff for an open set U⊂Y, f^-1(U) is open in X (also works for closed)

when are two metrics (on the same set) topologically equivalent

when the two metric spaces have exactly the same open sets

when are two metrics (on the same set) Lipschitz equivalent

if ∃ 0 < c ≤ C st

cd₁(x,y) ≤ d₂(x,y) ≤ Cd₁(x,y)

what can we say about the cty of f: X→Y and g: Y→X, and the topologically equivalent metrics d₁,d₂ on X
what about if the metrics are Lipschitz equivalent?

• f is cts from (X, d₁) into (Y, d_Y) iff its cts from (X, d₂) into (Y,d_Y)

• similarly for g

• same results hold for Lipschitz equivalence and Lipschitz cts

what is the relationship between Lipschitz and topological equivalence

Lipschitz is stronger ie Lipschitz implies topological, but not the other way round

what is the relationship between equivalent norms and equivalent metrics

two norms will induce equivalent metrics iff they are equivalent

define an isometry between X and Y

f: X→Y st f is a bijection, and distance is preserved

define a homeomorphism between X and Y

f: X→Y st f is a bijection, and both f and f^-1 are cts