Title: Theory / Unimportant / Math / Analysis / Real Pugh/{ / 4 - Function Spaces
Tags: fromMath mathToDef complex récupéré examplesSwitchExample1 automaticImplies closed counterExample hasSeenConjDef

Name: A sequence of function which converge pointwise to \(f\)
Definition: A sequence of function \((f_n)\) such that, for all \(x\), \(\lim_{n\to\infty}f_n(x)=f(x)\).
Page: 211
Chapter: 4
Section: 1
Variables: \((f_n)\) a sequence of function

Name: \((f_n)\) converges uniformly to \(f\)
Definition: For all \(\epsilon>0\), there exists \(N\in\mathbb N\) such that for all \(n>N\), \(|f_n(x)-f(x)|<\epsilon\) for each \(x\in[a,b]\).
Page: 211
Chapter: 4
Section: 1
Variables: \((f_n):[a,b]\to\mathbb R\) a sequence converging to \(f:[a,b]\to\mathbb R\).
Notation: \(\text{unif lim}_{n\to\infty}f_n=f\)
Extra: "b) "?
Notation2: \(f_n\rightrightarrows f\)

Name: The set of bounded real function on \([a,b]\).
Page: 214, 216
Chapter: 4
Section: 1
Variables: \(a<b\) two reals.
Notation: \(C_b([a,b],\mathbb R)\)
Index: 3
Kind: Theo
Typ: A complete metric space

Name: Sup norm of \(f\)
Definition: [$]\text{sup}\{\abs{f(x)}:x\in [a,b]\}[/$]
Page: 214
Chapter: 4
Section: 1
Variables: \(f\in C_b([a,b],\mathbb R)\).
Notation: [$]\norm{f}[/$]

Question: \(C^0([a,b],\mathbb R)\) is
Answer: a complete metric space
Page: 217
Chapter: 4
Section: 1
Kind: Corollary

Name: A sequence of function which converge absolutly
Definition: A sequence of function \((f_n)\in C_b([a,b],\mathbb R)\) such that [$]\sum_{i=0}^n\abs{f_k(x)}[/$] converges for each \(x\in[a,b]\).
Page: 217
Chapter: 4
Section: 1
Variables: \((f_n):[a,b]\to\mathbb R\) a sequence of function.

Name: A series of real functions which converge uniformly
Definition: A serie \((f_n):[a,b]\to\mathbb R\) such that the sequences \(\sum_{i=0}^n f_k(x)\) converge uniformly, for each \(x\in[a,b]\).
Page: 217
Chapter: 4
Section: 1

Name: Closed bounded non compact sets in \(C^0\), hence a sequence without accumulation point
Page: 223
Chapter: 4
Section: 3
Example: In the closed unit ball \(\mathcal B=\{f\in C^0([0,1],\mathbb R): ||f||\le 1\}\), all subsequence of \(x^n\) converges to \(\mathcal X_{\{1\} }\).

Name: A pointwise equicontinuous set of function
Definition: for each \(\epsilon>0\),
Page: 224, https://en.wikipedia.org/wiki/Equicontinuity
Chapter: 4
Section: 3
Variables: \(X,Y\) metric spaces,
Definition2: for each \(x_0\in X\)
Definition3: there exists \(\delta_{x_0,\epsilon}>0\) such that
Intuition: A set of functions equicontinuous at each \(x\in X\)
Definition4: for each \(x\in X\) with \(|x| <\delta_{x_0,\epsilon}\)
Definition5: for each \(f\in\mathcal F\)
DefType: A class \(\mathcal F\subseteq C^0(X,Y)\) such that
Definition6: \(|f(x)-f(x_0)|<\delta\).

Name: An equicontinuous set of function at \(x_0\)
Definition: for each \(\epsilon{\gt}0\),
Page: 224, https://en.wikipedia.org/wiki/Equicontinuity
Chapter: 4
Section: 3
Variables: \(X,Y\) metric spaces,
\(x_0\in X\)
Definition2: there exists \(\delta_{\epsilon}>0\) such that
Definition3: for each \(f\in\mathcal F\)
Definition4: for each \(x\in X\) with \(|x-x_0| <\delta_\epsilon\):
Definition5: \(|f(x)-f(x_0)|<\delta\).
DefType: A class \(\mathcal F\subseteq C^0(X,Y)\) such that

Name: An equicontinuous set of function
Definition: for each \(\epsilon>0\),
Page: 224, https://en.wikipedia.org/wiki/Equicontinuity
Chapter: 4
Section: 3
Definition2: there exists \(\delta_{\epsilon}>0\) such that
Conjdef: such that
Definition3: for each \(f\in\mathcal F\)
Name2: An uniformly equicontinuous set of function
Definition4: for each \(x,x'\in X\) with \(|x-x'| <\delta_{\epsilon}\):
Definition5: \(|f(x)-f(x')|<\epsilon\).
DefType: A class \(\mathcal F\subseteq C^0(X,Y)\)

Definition: \(C^0(M,\mathbb R)\).
Page: 234
Chapter: 4
Variables: \(M\) a compact metric space
Notation: \(C^0M\)

Name: A class of function which vanish at \(p\)
Definition: A class of functions whose elements \(f\) are such that \(f(p)=0\).
Page: 234
Chapter: 4
Variables: \(M\) a compact metric space
\(p\in M\).

Name: A class of functions which separate points
Definition: A class of functions such that for all \(p_1,p_2\in M\), there exists \(f\in\mathcal A\) such that \(f(p_1)\ne f(p_2)\).
Page: 234
Chapter: 4
Variables: \(M\) a compact metric space. Let \(\mathcal A\subseteq C^0M\).

Name: A function algebra
Definition: A set of function closed under addition, scalar multiplication and function multiplication.
Page: 234
Chapter: 4
Variables: \(M\) a compact metric space
Example: The set of polynomial.
Extra: type ?

Question: \(f\) can be uniformly approximated by
Answer: trigonometric polynomials.
Page: 238
Chapter: 4
Variables: \(f:\mathbb R\to\mathbb R\)
Index: 23
Kind: Cor
Assuming: \(f\) a \(2\pi\)-periodic continuous function

Name: A trigonometric polynomial
Definition: A function of the form \(a_0+\sum_{k=1}^n a_k \cos kx+\sum_{k=1}^n b_k\sin kx\).
Page: 238
Chapter: 4
Index: 23
Kind: Cor

Name: The class of trigonometric polynomials.
Page: 238
Chapter: 4
Notation: \(\mathcal T\)

Name: A fixed point of \(f\)
Definition: \(p\in M\) such that \(f(p)=p\).
Page: 240
Chapter: 4
Section: 5
Variables: \(M\) a compact metric space, \(f:M\to M\) and \(p\in M\).

Name: A contraction
Definition: \(f:M\to M\) such that \(d(f(x),f(y))\le k\ d(x,y)\) for some \(k<1\).
Page: 240
Chapter: 4
Section: 5
Variables: \(M\) a compact metric space

Smaller0: \(B^m\) (more generally compact)
Greater0: a fixed point \(p\in X\)
Page: 240
Chapter: 4
Section: 5
Variables: \(m\in\mathbb N\)
\(X\subseteq\mathbb R^m\)
\(f:X\to X\)
Index: 24
Prefix Greater: \(f\) has
Assuming: \(f\) is continuous
Name: Brouwer Fixed-Point Theorem
Extra: compact is enough
Prefix Smaller: \(X\) is
French: Théoreme du point fixe de Brouwer

Name: The closed unit ball in \(\mathbb R^m\)
Page: 240
Chapter: 4
Section: 5
Variables: \(m\in\mathbb N\).
Notation: \(B^m\)
Index: 24

Smaller0: \(\sigma\)
Greater0: \(I=[x-\sigma,x+\sigma]\)
Page: 249
Chapter: 4
Section: 6
Variables: \(f(x+h)=\sum c_k h^k\) with radius of convergence \(R>\sigma\).
Index: 27
Kind: Theorem
Hide middle: yes
Prefix Greater: \(f\) has bounded derivative growth rate of

Smaller0: \(f\) is analytic
Greater0: \(f\) has locally bounded derivative growth rate
Page: 250, 251
Chapter: 4
Section: 6
Variables: \(f:\mathbb C\to\mathbb C\) smooth
Index: 28, 29
Kind: Theo, cor
Smaller to greater: iff
Name: Analyticity Theorem
Greatest: \(f\) has a uniformly bounded derivatives

Name: Continuous real function that has a derivative at no point
Page: 254
Chapter: 4
Section: 7
Example: \[\sum\sigma_k(x)\]
Where \[\sigma_0(x)=\left\{ \begin{array}{ll} x-2n &\text{if }2n\le x\le 2n+1\\ 2n+2-x&\text{if }2n+1\le x\le 2n+2 \end{array}\right.\]
and \(\sigma_k(x)=\left(\frac 34\right)^k\sigma_0(4^kx)\).

Question: The class of open-dense sets is closed under
Answer: intersection.
Page: 256
Chapter: 4
Section: 7

Name: Generic subset of a metric space
Definition: A subset which includes a thick subset of \(M\).
Page: 256
Chapter: 4
Section: 7
Variables: \(M\) be a metric space
\(P\subseteq M\) a property.

Smaller0: The generic \(f\in C^0([a,b],\mathbb R)\) is
Greater0: at no points
Page: 256
Chapter: 4
Section: 7
Index: 33
Kind: Theo
Smaller to greater: iff
Hide middle: yes
Prefix Greater: differentiable
Greater1: not monotone on any subinterval of \([a,b]\)
Connect Greater0: and

Question: The generic \(f\in C^0([a,b],\mathbb R)\) is monotone on
Answer: no subinterval of \([a,b]\).
Page: 256
Chapter: 4
Section: 7
Index: 33
Kind: Theo

Question: \(\mathbb R\setminus\mathbb Z\) is
Answer: a not thick subset of \(\mathbb R^2\).
Page: 257
Chapter: 4
Section: 7

Name: A \(\sigma\)-compact over \(M\)
Definition: A countable union of compact sets of \(M\)
Page: 262, https://en.m.wikipedia.org/wiki/Σ-compact_space
Chapter: 4
Section: 8
Variables: \(M\) a topological space.
Example: Compact spaces
Example2: \(\mathbb R^n\)
Example3: Locally compact at one point topological group
Counterexample: lower limit topology
Example4: Hemicompact
Counterexample2: countable complement topology
Counterexample3: \(\mathbb R^\omega\)

Smaller0: \(M\) complete
Greater0: \(S\) is dense in \(M\)
Page: 256
Chapter: 4
Section: 7
Variables: \(M\) a metric space
\(S\subseteq M\).
Index: 32
Kind: Theorem
Hide middle: yes
Smaller1: \(S\) is a thick subset
Name: Baire's theorem

Smaller0: \(\mathcal E\) is compact
Greater0: \(\mathcal E\) is closed
Page: 228
Chapter: 4
Section: 3
Variables: \(\mathcal E\subseteq C^0([a,b],\mathbb R)\).
Index: 18
Kind: Theo
Smaller to greater: iff
Greater1: \(\mathcal E\) is bounded
Greater2: \(\mathcal E\) is equicontinuous

Smaller0: \((f_n)\rightrightarrows f\)
Greater0: \(f\) is continuous at \(x\)
Page: 213
Chapter: 4
Section: 1
Variables: \((f_n):[a,b]\to\mathbb R\) a sequence of function with limit \(f\).
Kind: Theo
Smaller1: the \(f_n\)'s are continuous at \(x\)

Name: Growing steeple
Definition: \(f_n(x)=\left\{\begin{array}{ll}n^2x & \text{if }0\le x\le \frac 1n\\2n-n^2x & \text{if }\frac 1n\le x\le\frac2n\\0 & \text{if }\frac2n\le x\le 1\end{array}\right.\)
Page: 214
Chapter: 4
Section: 1

Name: Sequence of continuous function, pointwise converging, on a compact, whose limit is continuous but not uniformly continuous
Page: 214
Chapter: 4
Section: 1
Example: Growing steeple

Smaller0: \((f_n)\) converges for the sup norm
Greater0: \((f_n)\) converges uniformly
Page: 216
Chapter: 4
Section: 1
Variables: \((f_n)\in C_b([a,b],\mathbb R)\).
Index: 2
Kind: Theorem
Smaller to greater: iff
Hide middle: yes

Smaller0: \(\sum M_k\) is a convergent series of constant and if \(||f_k||\le M_k\) for all \(k\)
Greater0: \(\sum f_k\) converges uniformly and absolutely
Page: 217
Chapter: 4
Section: 1
Variables: \((M_k)\in \mathbb R\)
\((f_k)\in C_b([a,b],\mathbb R)\).
Index: 5
Kind: doublon name
Hide middle: yes
Name: Weiersrtrass M-test
Suffix Smaller: ,

Smaller0: \((f_n)\) converges uniformly
Greater0: [$]\lim_{n\to\infty}\int_a^b f_n(x)\dd x=\int_a^b\text{lim}_{n\to\infty}f_n(x)\dd x[/$]
Page: 218
Chapter: 4
Section: 1
Variables: \((f_n):[a,b]\to\mathbb R\) a sequence of function.
Index: 6
Kind: Theo
Hide middle: yes

Smaller0: \(f\) is a differentiable surjection, and \(f'(x)\ne 0\) for all \(x\in(a,b)\)
Greater0: \(f\) is a homeomorphism
Chapter: 3
Variables: \(f:(a,b)\to (c,d)\).
Hide middle: yes

Smaller0: \((f_n)\) converges uniformly
Greater0: [$]\lim_{n\to\infty}\int_a^xf_n(x)\dd x\rightrightarrows\int_a^x\text{lim}_{n\to\infty}f_n(x)\dd x[/$]
Page: 218
Chapter: 4
Section: 1
Variables: \((f_n):[a,b]\to\mathbb R\) a sequence of function.
Index: 7
Kind: Cor
Hide middle: yes

Smaller0: \(\sum f_k\) is a uniformly convergent series of integrable function
Greater0: [$]\int_a^b\sum f_k(x)\dd x=\sum\int_a^b f_k(x)\dd x[/$]
Page: 219
Chapter: 4
Section: 1
Variables: \((f_n):[a,b]\to\mathbb R\) a sequence of function.
Index: 8
Hide middle: yes

Smaller0: the \(f_n\) are differentiable
Greater0: \(\text{lim}_{n\rightarrow\infty}f_n\) is differentiable
Page: 219
Chapter: 4
Section: 1
Variables: \((f_n):[a,b]\to\mathbb R\) a sequence of function.
Index: 9
Kind: Theo
Smaller1: \((f_n)\) and \((f'_n)\) converges uniformly
Connect Smaller0: , and

Smaller0: the \(f_n\) are differentiable, and \(\sum f_n\) and \(\sum f'_n\) are uniformly convergent series
Greater0: \(\left(\sum f_k(x)\right)'=\sum f'_k(x)\)
Page: 220
Chapter: 4
Section: 1
Variables: \((f_n):[a,b]\to\mathbb R\) a sequence of function.
Index: 10
Kind: Theo
Hide middle: yes

Question: Then \(f\), on \([-r,r]\) have the property
Answer: to converge uniformly and absolutely.
Page: 221
Chapter: 4
Section: 2
Variables: \(f(x)=\sum c_kx^k\) with radius of convergence \(R\) and \(r<R\).
Index: 11
Kind: Theo

Name: A non-differentiable uniform limit of differentiable function
Page: 220
Chapter: 4
Section: 2
Variables: \((f_n):\mathbb R\to\mathbb R\).
Example: \(f_n(x)=\sqrt{x^2+\frac 1n}\) converges to \(f(x)=|x|\).

Name: A vector Ordinary Differential Equation (ODE) on \(U\)
Definition: \(m\) simultaneous scalar equations \(x'_i=f_i(x_1,\dots,x_m)\) with the \(f_i:U\to \mathbb R\).
Page: 242
Chapter: 4
Section: 5
Variables: \(U\subseteq \mathbb R^m\) open.

Name: initial condition
Definition: Some fixed values for \(x_1(0),\dots,x_m(0)\).
Page: 242
Chapter: 4
Section: 5
Variables: \(U\subseteq \mathbb R^m\) open.
Let \(n\in\mathbb N\),
Let \(x'_i=f_i(x_1,\dots,x_m)\) with the \(f_i:U\to \mathbb R\) an ODE
Context: Ordinary differential equation

Name: The solution of this ODE
Definition: Some \(x_1(t),\dots,x_m(t)\) such that \(\frac{dx_i(t)}{dt}=f_i(x_1(t),\dots,x_m(t))\) for all \(i\)
Page: 242
Chapter: 4
Section: 5
Variables: \(U\subseteq \mathbb R^m\) open.
Let \(n\in\mathbb N\),
Let \(x'_i=f_i(x_1,\dots,x_m)\) with the \(f_i:U\to \mathbb R\) an ODE
Let \(x_1(0),\dots,x_m(0)\) initial conditions

Smaller0: \(f'(x)\)
Greater0: \(\sum c_{k+1}(k+1)x^k\)
Page: 221
Chapter: 4
Section: 2
Variables: \(f(x)=\sum c_kx^k\) with radius of convergence \(R\).
On \((-R,R)\),
Index: 12
Kind: Theo
Smaller to greater: =
Hide middle: yes

Definition: \(\sum\frac{(-1)^k x^{2k+1} }{(2k+1)!}\)
Page: 222
Chapter: 4
Section: 2
Notation: \(\sin(x)\)

Question: \(\cos'=\)
Answer: \(-\sin\)
Page: 223
Chapter: 4
Section: 2

Question: \(\sin'=\)
Answer: \(\cos\)
Page: 223
Chapter: 4
Section: 2

Smaller0: \(\log(1+x)\) for \(|x|<1\)
Greater0: \(\sum \frac{(-1)^{k+1}x^k}{k}\)
Page: 223
Chapter: 4
Section: 2
Smaller to greater: is equal to
Hide middle: yes

Smaller0: there exists a dense subset \(S\) of the domain such that \((f_k)\) is bounded equicontinuous on \(S\)
Greater0: \((f_k)\) is uniformly convergent on the whole domain
Page: 224
Chapter: 4
Section: 3
Variables: \((f_k)\in C^0([a,b],\mathbb R)\).
Hide middle: yes
Name: Arzelà-Ascoli theorem

Smaller0: the \(f_n\) are differentiable, \((f'_n)\) is uniformly bounded and \(f_n(x)\) is bounded for some \(x\in[a,b]\)
Greater0: \((f_n)\) has a subsequence that converges uniformly on \([a,b]\)
Page: 227
Chapter: 4
Section: 3
Variables: \((f_n):[a,b]\to\mathbb R\)
Index: 17
Kind: Cor
Hide middle: yes

Question: A Property of the the set of polynomials in \(C^0([a,b],\mathbb R)\):
Answer: it is dense.
Page: 228
Chapter: 4
Index: 19
Kind: Theo
Name: Weierstrass approximation theorem

Name: The Bernstein basis polynomials of degree n
Definition: \[{k\choose ne}x^k(1-x)^{n-k}\].
Page: 229
Chapter: 4
Index: 19
Kind: Theo
Extra: ne ?

Question: Example of sequence of approximation of \(f\) by polynomials
Answer: \(p_n(x)=\sum_{k=0}^n{k\choose n} f(\frac kn) x^k(1-x)^{n-k}\)
Page: 229
Chapter: 4
Variables: \(f:[0,1]\to\mathbb R\),
Index: 19
Kind: Theo
Extra: + (page pas trouvé)

Smaller0: a function algebra that vanishes nowehere and separates points
Greater0: dense in \(C^0M\)
Page: 234
Chapter: 4
Variables: \(M\) a compact metric space
\(\mathcal A\subseteq C^0M\).
Index: 20
Kind: Theo
Hide middle: yes
Prefix Greater: \(\mathcal A\) is
Name: Stone-Weierstrass Theorem
Suffix Smaller: in \(\mathcal A\)

Name: Banach Contraction Principle
Definition: \(f\) has a unique fixed-point \(p\) and for any \(x_0\in M\), the sequence \((x_n)\) defined by \(x_{n+1}=f(x_n)\) converges to \(p\) as \(n\to \infty\).
Page: 240
Chapter: 4
Section: 5
Variables: \(M\) a compact metric space
\(f:M\to M\).
Index: 24
Assuming: \(f\) is a contraction and \(M\) is complete

Question: The sum of the Bernstein basis polynomials of degree \(n\) is
Answer: 1
Page: 229
Chapter: 4
Variables: \(n\in\mathbb N\).
Index: 19
Kind: Theo

Smaller0: \(C^{ \omega }\)
Greater0: \( C^{ \infty}\)
Page: 222
Chapter: 4
Section: 2
Index: 13
Kind: Theorem
Smaller to greater: \(\subseteq\)
Greater to smaller: \(\supseteq\)

Question: There exists a curve
Answer: an \(F\)-trajectory \(\gamma(t)\) in \(U\) through \(p\).
Page: 244
Chapter: 4
Section: 5
Variables: \(U\) be an open subset of \(\mathbb R^m\) and
\(F:\mathbb R^m\to\mathbb R^m\).
\(p\in U\)
Index: 25
Kind: Theo
Name: Picards's Theorem
Extra: locally unique ?

Name: An \(F\)-trajectory in \(U\) through \(p\)
Definition: A curve \(\gamma:(a,b)\to U\) such that \(a<0<b\) and for all \(t\in (a,b)\):
\(\gamma'(t)=F(\gamma (t))\text{ and } \gamma (0)=p\).
Page: 244
Chapter: 4
Section: 5
Variables: \(U\) be an open subset of \(\mathbb R^m\) and
\(F:\mathbb R^m\to\mathbb R^m\).
\(p\in U\)
Index: 25
Kind: Theo
Extra: locally unique ?
Typ: locally unique.

Question: Let \(\alpha\) the derivative growth rate of \(f\) on \(I\) and \(R\) the radius of convergence of \(f\).
Then
Answer: \(1\le R\alpha\).
Page: 248
Chapter: 4
Section: 6
Variables: \(f:(a,b)\to\mathbb R\) analytic.
\(I\subseteq(a,b)\) an interval.

Name: The derivative growth rate of \(f\) on \(I\)
Definition: \(\text{lim sup}_{r\to\infty}\sqrt[r]{\frac{\text{max}_{t\in I}|f^{(r)}(t)|}{r!} }\).
Page: 248
Chapter: 4
Section: 6
Variables: \(f:(a,b)\to\mathbb R\) analytic. Let \(I\subseteq(a,b)\) an interval.
Extra: vérifier

Smaller0: \(\alpha\sigma<1\)
Greater0: the Taylor series converges uniformly on \(I\)
Page: 249
Chapter: 4
Section: 6
Variables: \(f\) be an analytic function of derivative growth rate \(\alpha\) on \(I=[x-\sigma, x+\sigma]\).
Index: 26
Kind: Theo
Hide middle: yes

Question: Proporty of sinus
Answer: odd.

Smaller0: the power series has radius of convergence \(R\)
Greater0: \((-R,R)\)
Page: 251
Chapter: 4
Section: 6
Variables: \(f(x)=\sum c_k x^k\).
Index: 30
Hide middle: yes
Prefix Greater: \(f\) is analytic on
Name: Taylor's Theorem

Question: \(\sin(0)=\)
Answer: \(1\)

Question: \(\cos(0)=\)
Answer: \(0\)

Question: \(e^{i\theta}=\)
Answer: \(\cos(\theta)+i\sin(\theta) \)
Page: 252
Chapter: 4
Section: 7
Variables: \(\theta\in\mathbb R\)

Name: A thick subset of \(M\)
Definition: A countable intersection of open-dense set.
Page: 256, 257
Chapter: 4
Section: 7
Variables: \(M\) a topological space
\(G\subseteq M\).
Example: \(M\),
Name2: A residual subset of \(M\)
Example2: \(\mathbb R\setminus\mathbb Z\) in \(\mathbb R\)

Name: A thin subset of \(M\)
Definition: A not thick subset of \(M\).
Page: 256, 257
Chapter: 4, 4
Section: 7, 7
Variables: \(M\) a topological space.
Definition2: countable union of nowhere dense set
Example: \(\emptyset\)
Name2: A meager subset of \(M\)
Example2: The Cantor space in \(\mathbb R\)
Example3: \(\mathbb Q\) in \(\mathbb R\) (and countable sets ?)

Question: The generic square matrix has determinant
Answer: not 0.
Page: 257
Chapter: 4
Section: 7

Question: The generic linear transformation \(\mathbb R^m\to\mathbb R^m\) is an
Answer: isomorphism.
Page: 257
Chapter: 4
Section: 7
Variables: \(m\in\mathbb N\).

Question: The generic linear transformation \(\mathbb R^m\to\mathbb R^{m-k}\) is
Answer: onto.
Page: 257
Chapter: 4
Section: 7
Variables: \(m,k\in\mathbb N\).

Question: The generic linear transformation \(\mathbb R^m\to\mathbb R^{m+k}\) is
Answer: one-to-one.
Page: 257
Chapter: 4
Section: 7
Variables: \(m,k\in\mathbb N\).

Question: The generic pair of lines of \(\mathbb R^3\) are
Answer: skew (non parallel and disjoint)
Page: 257
Chapter: 4
Section: 7

Question: The number of distinct roots of the generic \(n^{\text{th}}\)-degree polynomial is
Answer: \(n\).
Page: 257
Chapter: 4
Section: 7

Smaller0: \(S\) is a nonempty,
Greater0: \(S\) is not thin
Page: 256
Chapter: 4
Section: 7
Variables: \(M\) a topological space and \(S\subseteq M\).
Index: 32
Kind: Theo
Hide middle: yes
Smaller1: \(S\) is a complete metric space
Name: Baire's theorem

Smaller0: \(M\) is a complete nonempty metric space
Greater0: \(S\) is not thin
Page: 258
Chapter: 4
Section: 7
Variables: \(S\subseteq M\).
Index: 34
Kind: Cor
Hide middle: yes
Smaller1: \(S\) is thick

Smaller0: \((f_n)\) is uniformly equicontinuous
Greater0: \((f_n)\) has a uniformly convergent subsequence
Page: 261
Chapter: 4
Section: 8
Variables: \(X,Y\) be metric spaces
\((f_n):X\to Y\)
Index: 37
Kind: Theo
Hide middle: yes
Smaller1: \(X\) is compact
Smaller2: for each \(x\in X\) the sequence \((f_n(x))\) lies in a compact subset of \(Y\)

Smaller0: \((f_n)\) converges uniformly
Greater0: \((f_n)\) converges for the \(d\)-metric
Page: 261
Chapter: 4
Section: 8
Variables: \(X,Y\) be metric spaces
\(\mathcal F\) be the set of functions from \(X\) to \(Y\) with metric space \(d(f,g)=\text{sup}_{x\in X}\frac{d_Y(fx,gx)}{1+d_Y(fx,gx)}\)
\((f_n)\in\mathcal F\).
Index: 35a
Kind: Theorem
Smaller to greater: iff

Smaller0: \(Y\) is complete
Greater0: \(\mathcal F\) is complete
Page: 261
Chapter: 4
Section: 8
Variables: \(X,Y\) be metric spaces.
\(\mathcal F\) be the set of functions from \(X\) to \(Y\) with metric space \(d(f,g)=\text{sup}_{x\in X}\frac{d_Y(fx,gx)}{1+d_Y(fx,gx)}\).
Index: 35 b
Kind: Theo
Hide middle: yes
Extra: of continuous function ?

Question: Topologically, \(\mathcal F_{\text{b}}\) is
Answer: closed.
Page: 261
Chapter: 4
Section: 8
Variables: \(X\) be metric
\(Y\) be complete
\(\mathcal F\) be the set of bounded functions from \(X\) to \(Y\) with metric space \(d(f,g)=\text{sup}_{x\in X}\frac{d_Y(fx,gx)}{1+d_Y(fx,gx)}\)
\(\mathcal F_{\text{b}}\subseteq\mathcal F\), the set of bounded function of \(\mathcal F\)
Index: 35c
Kind: Theo

Question: Topologically, \(C^0(X,Y)\):
Answer: it is closed.
Page: 261
Chapter: 4
Section: 8
Variables: \(X,Y\) be metric spaces
\(\mathcal F\) be the set of functions from \(X\) to \(Y\) with metric space \(d(f,g)=\text{sup}_{x\in X}\frac{d_Y(fx,gx)}{1+d_Y(fx,gx)}\).
Kind: Theo

Smaller0: \(X\) is compact
Greater0: uniformly equicontinuous
Page: 261
Chapter: 4
Section: 8
Variables: \(X,Y\) be metric spaces
\((f_n):X\to Y\).
Index: 36
Kind: Theo
Hide middle: yes
Prefix Greater: \(f_n\) is
Smaller1: \((f_n)\) is pointwise equicontinuous

Smaller0: \((f_n)\) is pointwise equicontinuous
Greater0: \((f_n)\) has a subsequence that converges uniformly to a limit on each compact subset of \(X\)
Page: 262
Chapter: 4
Section: 8
Variables: \(X,Y\) metric spaces
\((f_n):X\to Y\).
Index: 38
Kind: Theo
Hide middle: yes
Smaller1: \(X\) is \(\sigma-\)compact
Smaller2: for each \(x\in X\) the sequence \((f_n(x))\) lies in a compact subset of \(Y\)

Smaller0: \((f_n)\) is pointwise equicontinuous
Greater0: \(f_n\) has a subsequence that converges uniformly on every compact subset of \(\mathbb R\)
Page: 262
Chapter: 4
Section: 8
Variables: \((f_n):\mathbb R\to \mathbb R\).
Index: 39
Kind: Cor
Smaller1: for some \(x_0\), \(f_n(x_0)\) is bounded

Name: -compact topological space
Page: https://en.wikipedia.org/wiki/%CE%A3-compact_space
Under: product
Condition: finite
Under2: intersection ?
Under3: union?