10.1 Definitions[2CC]
A metric space is a pair \((X,d)\) where \(X\) is a set (nonempty) with associated distance \(d\).
[0MS] A distance is a function \(d:X× X→ [0,∞)\) that enjoys the following properties:
\(d(x,x)=0\);
(separation property) if \(d(x,y)=0\) then \(x=y\);
(symmetry) \(d(x,y)=d(y,x)\) for each \(x,y∈ X\);
(triangle inequality) \(d(x,z)≤ d(x,y)+d(y,z)\) for each \(x,y,z∈ X\).
[0MT] Given a sequence \((x_ n)_ n⊆ X\) and \(x∈ X\),
we will say that ”\((x_ n)_ n\) converges to \(x\)” if \(\lim _ n d(x_ n,x)=0\); we will also write \(x_ n→_ n x\) to indicate that the sequence converges to \(x\).
We will say that ”\((x_ n)_ n\) is a Cauchy sequence” if
\[ ∀ \varepsilon {\gt}0~ ~ ∃ N∈ℕ~ ,~ ∀ n,m≥ N~ ~ d(x_ n,x_ m){\lt}\varepsilon ~ ~ . \]
[2C1]To any given set \(X\) we may associate the discrete distance
The induced topology is the discrete topology where every subset of \(X\) is an open set.
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[0MV]Prove that a converging sequence \((x_ n)_ n⊆ X\) is Cauchy.
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[0MW]Given a sequence \((x_ n)_ n⊆ X\) show that, if it converges to \(x\) and converges to \(y\), then \(x=y\).
This result is known as Theorem of the uniqueness of the limit.
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[0MX]We generalize the definition of metric space assuming that \(d:X→[0,∞]\) (the other axioms are the same). Show that the relation \(x∼ y\) defined by
\[ x∼ y\iff d(x,y){\lt}∞ \]is an equivalence relation, and that equivalence classes are open, and therefore are disconnected from each other.
Hidden solution: [UNACCESSIBLE UUID ’0MY’]
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[0MZ] Given \(f,g\) continuous functions on \(ℝ\), we define
\[ d(f,g)=\sup _{x∈ℝ}|f(x)-g(x)|\ . \]Prove that \(d\) is a distance on \(X=C(ℝ)\), in the extended sense of the exercise 3.
Let \(f∼ g\iff d(f,g){\lt}∞\) as before, show that the family of equivalence classes \(\frac X∼\) has the cardinality of the continuum.
Hidden solution: [UNACCESSIBLE UUID ’0N0’]
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[0N1] Prerequisites:15.Note:See also eserc. 2. Suppose \(𝜑:[0,∞)→[0,∞)\) is monotonic weakly increasing and subadditive, i.e. \(𝜑(t)+𝜑(s)≥ 𝜑(t+s)\) for each \(t,s≥ 0\); and suppose that \(𝜑(x)=0\) if and only if \(x=0\).
Then \(𝜑◦ d\) is again a distance. Examples: \(𝜑(t)=\sqrt t\), \(𝜑(t)=t/(1+t)\), \(𝜑(t)=\arctan (t)\), \(𝜑(t)=\min \{ t,1\} \).
Moreover show that if \(𝜑\) is continuous in zero then the associated topology is the same. 1 Hidden solution: [UNACCESSIBLE UUID ’0N2’]
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[0N3] If \((x_ n)_ n⊂ X\) is a sequence and \(x∈ X\), show that \(\lim _{n→∞} x_ n=x\) if and only if, for each sub–sequence \(n_ k\) there exists a sub–sub–sequence \(n_{k_ h}\) such that \(\lim _{h→∞} x_{n_{k_ h}}=x\). Hidden solution: [UNACCESSIBLE UUID ’0N4’]
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[0N5]A sequence \((x_ n)⊂ X\) is a Cauchy sequence if and only if
\[ \lim _{N→∞}\sup \{ d(x_ n,x_ m) : n≥ N, m≥ N\} =0~ ~ . \]- E279
[0N6] A sequence \((x_ n)⊂ X\) is a Cauchy sequence if and only if there exists a sequence \(\varepsilon _ n\) with \(\varepsilon _ n≥ 0\) and \(\varepsilon _ n→_ n 0\) such that, for every \(n\) and every \(m≥ n\), we have \(d(x_ n,x_ m)≤ \varepsilon _ n\).
Hidden solution: [UNACCESSIBLE UUID ’0N7’]
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[0N8] If \((x_ n)⊂ X\) is a Cauchy sequence and there exists \(x\) and a subsequence \(n_ m\) such that \(\lim _{m→∞} x_{n_ m}=x\) then \(\lim _{n→∞} x_{n}=x\).
Hidden solution: [UNACCESSIBLE UUID ’0N9’]
This ”lemma” is used in some important proofs, e.g. to show that a compact metric space is also complete.
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[0NC] Let \(\varepsilon _ n{\gt}0\) be an infinitesimal decreasing sequence. If \((x_ n)⊂ X\) is a Cauchy sequence, there exists a subsequence \(n_ k\) such that
\[ ∀ k ∈ℕ,~ ∀ h ∈ℕ,~ h{\gt}k ⇒ d(x_{n_ k},x_{n_ h})≤ \varepsilon _ k~ ~ . \]Hidden solution: [UNACCESSIBLE UUID ’0ND’] This property is often used by choosing \(\varepsilon _ n=2^{-n}\), or other sequence whose series converges.
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[0NF] Let \((x_ n)_ n\) be a sequence such that \(∑_{n=1}^∞ d(x_ n,x_{n+1}) {\lt} ∞\): prove that it is a Cauchy sequence.
Compare this exercise, the previous 10 in case \(∑_ n \varepsilon _ n{\lt}∞\), and exercise 9.
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[0NG]If \((x_ n)⊂ X\) is a Cauchy sequence, \((y_ n)⊂ X\) is another sequence, and \(d(x_ n,y_ n)→_ n 0\), then \((y_ n)⊂ X\) is a Cauchy sequence.
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[0NH] Given \((X,d)\) a metric space, show that \(d\) is continuous (as a function \(d:X× X→ℝ\)). You can actually show that it is Lipschitz, by associating to \(X× X\) the distance
\[ \hat d (x,y) = d(x_ 1,y_ 1) + d(x_ 2,y_ 2), \text{ for } x=(x_ 1,x_ 2),y=(y_ 1,y_ 2) \in X× X~ . \]Hidden solution: [UNACCESSIBLE UUID ’0NK’] [UNACCESSIBLE UUID ’0NJ’]
Let \(𝛼(x)\) be a continuous function on \(ℝ\), bounded and strictly positive. Given \(f,g\) continuous functions on \(ℝ\), we define
Prove that \(d\) is a distance on \(C(ℝ)\) and that \(\big(C(ℝ),d\big)\) is complete. Hidden solution: [UNACCESSIBLE UUID ’0NP’] [0NQ] Note:Exercise 2, written exam, 25 March 2017.
Show that the following properties are equivalent for a metric space \(X\):
every sequence of elements of \(X\) admits a Cauchy subsequence;
The completion \(X^*\) of \(X\) is compact.
Hidden solution: [UNACCESSIBLE UUID ’0NR’]
[UNACCESSIBLE UUID ’0NS’] [UNACCESSIBLE UUID ’0NT’] [UNACCESSIBLE UUID ’0NV’]