formal
stringlengths 51
351
| informal
stringlengths 3
474
|
---|---|
theorem exercise_1_1b
(x : β)
(y : β)
(h : y β 0)
: ( irrational x ) -> irrational ( x * y ) := | If $r$ is rational $(r \neq 0)$ and $x$ is irrational, prove that $rx$ is irrational. |
theorem exercise_1_4
(Ξ± : Type*) [partial_order Ξ±]
(s : set Ξ±)
(x y : Ξ±)
(hβ : set.nonempty s)
(hβ : x β lower_bounds s)
(hβ : y β upper_bounds s)
: x β€ y := | Let $E$ be a nonempty subset of an ordered set; suppose $\alpha$ is a lower bound of $E$ and $\beta$ is an upper bound of $E$. Prove that $\alpha \leq \beta$. |
theorem exercise_1_8 : Β¬ β (r : β β β β Prop), is_linear_order β r := | Prove that no order can be defined in the complex field that turns it into an ordered field. |
theorem exercise_1_12 (n : β) (f : β β β) :
abs (β i in finset.range n, f i) β€ β i in finset.range n, abs (f i) := | If $z_1, \ldots, z_n$ are complex, prove that $|z_1 + z_2 + \ldots + z_n| \leq |z_1| + |z_2| + \cdots + |z_n|$. |
theorem exercise_1_14
(z : β) (h : abs z = 1)
: (abs (1 + z)) ^ 2 + (abs (1 - z)) ^ 2 = 4 := | If $z$ is a complex number such that $|z|=1$, that is, such that $z \bar{z}=1$, compute $|1+z|^{2}+|1-z|^{2}$. |
theorem exercise_1_17
(n : β)
(x y : euclidean_space β (fin n)) -- R^n
: βx + yβ^2 + βx - yβ^2 = 2*βxβ^2 + 2*βyβ^2 := | Prove that $|\mathbf{x}+\mathbf{y}|^{2}+|\mathbf{x}-\mathbf{y}|^{2}=2|\mathbf{x}|^{2}+2|\mathbf{y}|^{2}$ if $\mathbf{x} \in R^{k}$ and $\mathbf{y} \in R^{k}$. |
theorem exercise_1_18b
: Β¬ β (x : β), β (y : β), y β 0 β§ x * y = 0 := | If $k = 1$ and $\mathbf{x} \in R^{k}$, prove that there does not exist $\mathbf{y} \in R^{k}$ such that $\mathbf{y} \neq 0$ but $\mathbf{x} \cdot \mathbf{y}=0$ |
theorem exercise_2_19a {X : Type*} [metric_space X]
(A B : set X) (hA : is_closed A) (hB : is_closed B) (hAB : disjoint A B) :
separated_nhds A B := | If $A$ and $B$ are disjoint closed sets in some metric space $X$, prove that they are separated. |
theorem exercise_2_25 {K : Type*} [metric_space K] [compact_space K] :
β (B : set (set K)), set.countable B β§ is_topological_basis B := | Prove that every compact metric space $K$ has a countable base. |
theorem exercise_2_27b (k : β) (E P : set (euclidean_space β (fin k)))
(hE : E.nonempty β§ Β¬ set.countable E)
(hP : P = {x | β U β π x, (P β© E).nonempty β§ Β¬ set.countable (P β© E)}) :
set.countable (E \ P) := | Suppose $E\subset\mathbb{R}^k$ is uncountable, and let $P$ be the set of condensation points of $E$. Prove that at most countably many points of $E$ are not in $P$. |
theorem exercise_2_29 (U : set β) (hU : is_open U) :
β (f : β β set β), (β n, β a b : β, f n = {x | a < x β§ x < b}) β§ (β n, f n β U) β§
(β n m, n β m β f n β© f m = β
) β§
U = β n, f n := | Prove that every open set in $\mathbb{R}$ is the union of an at most countable collection of disjoint segments. |
theorem exercise_3_2a
: tendsto (Ξ» (n : β), (sqrt (n^2 + n) - n)) at_top (π (1/2)) := | Prove that $\lim_{n \rightarrow \infty}\sqrt{n^2 + n} -n = 1/2$. |
theorem exercise_3_5 -- TODO fix
(a b : β β β)
(h : limsup a + limsup b β 0) :
limsup (Ξ» n, a n + b n) β€ limsup a + limsup b := | For any two real sequences $\left\{a_{n}\right\},\left\{b_{n}\right\}$, prove that $\limsup _{n \rightarrow \infty}\left(a_{n}+b_{n}\right) \leq \limsup _{n \rightarrow \infty} a_{n}+\limsup _{n \rightarrow \infty} b_{n},$ provided the sum on the right is not of the form $\infty-\infty$. |
theorem exercise_3_7
(a : β β β)
(h : β y, (tendsto (Ξ» n, (β i in (finset.range n), a i)) at_top (π y))) :
β y, tendsto (Ξ» n, (β i in (finset.range n), sqrt (a i) / n)) at_top (π y) := | Prove that the convergence of $\Sigma a_{n}$ implies the convergence of $\sum \frac{\sqrt{a_{n}}}{n}$ if $a_n\geq 0$. |
theorem exercise_3_13
(a b : β β β)
(ha : β y, (tendsto (Ξ» n, (β i in (finset.range n), |a i|)) at_top (π y)))
(hb : β y, (tendsto (Ξ» n, (β i in (finset.range n), |b i|)) at_top (π y))) :
β y, (tendsto (Ξ» n, (β i in (finset.range n),
Ξ» i, (β j in finset.range (i + 1), a j * b (i - j)))) at_top (π y)) := | Prove that the Cauchy product of two absolutely convergent series converges absolutely. |
theorem exercise_3_21
{X : Type*} [metric_space X] [complete_space X]
(E : β β set X)
(hE : β n, E n β E (n + 1))
(hE' : tendsto (Ξ» n, metric.diam (E n)) at_top (π 0)) :
β a, set.Inter E = {a} := | If $\left\{E_{n}\right\}$ is a sequence of closed nonempty and bounded sets in a complete metric space $X$, if $E_{n} \supset E_{n+1}$, and if $\lim _{n \rightarrow \infty} \operatorname{diam} E_{n}=0,$ then $\bigcap_{1}^{\infty} E_{n}$ consists of exactly one point. |
theorem exercise_4_1a
: β (f : β β β), (β (x : β), tendsto (Ξ» y, f(x + y) - f(x - y)) (π 0) (π 0)) β§ Β¬ continuous f := | Suppose $f$ is a real function defined on $\mathbb{R}$ which satisfies $\lim_{h \rightarrow 0} f(x + h) - f(x - h) = 0$ for every $x \in \mathbb{R}$. Show that $f$ does not need to be continuous. |
theorem exercise_4_3
{Ξ± : Type} [metric_space Ξ±]
(f : Ξ± β β) (h : continuous f) (z : set Ξ±) (g : z = fβ»ΒΉ' {0})
: is_closed z := | Let $f$ be a continuous real function on a metric space $X$. Let $Z(f)$ (the zero set of $f$ ) be the set of all $p \in X$ at which $f(p)=0$. Prove that $Z(f)$ is closed. |
theorem exercise_4_4b
{Ξ± : Type} [metric_space Ξ±]
{Ξ² : Type} [metric_space Ξ²]
(f g : Ξ± β Ξ²)
(s : set Ξ±)
(hβ : continuous f)
(hβ : continuous g)
(hβ : dense s)
(hβ : β x β s, f x = g x)
: f = g := | Let $f$ and $g$ be continuous mappings of a metric space $X$ into a metric space $Y$, and let $E$ be a dense subset of $X$. Prove that if $g(p) = f(p)$ for all $p \in P$ then $g(p) = f(p)$ for all $p \in X$. |
theorem exercise_4_5b
: β (E : set β) (f : β β β), (continuous_on f E) β§
(Β¬ β (g : β β β), continuous g β§ β x β E, f x = g x) := | Show that there exist a set $E \subset \mathbb{R}$ and a real continuous function $f$ defined on $E$, such that there does not exist a continuous real function $g$ on $\mathbb{R}$ such that $g(x)=f(x)$ for all $x \in E$. |
theorem exercise_4_8a
(E : set β) (f : β β β) (hf : uniform_continuous_on f E)
(hE : metric.bounded E) : metric.bounded (set.image f E) := | Let $f$ be a real uniformly continuous function on the bounded set $E$ in $R^{1}$. Prove that $f$ is bounded on $E$. |
theorem exercise_4_11a
{X : Type*} [metric_space X]
{Y : Type*} [metric_space Y]
(f : X β Y) (hf : uniform_continuous f)
(x : β β X) (hx : cauchy_seq x) :
cauchy_seq (Ξ» n, f (x n)) := | Suppose $f$ is a uniformly continuous mapping of a metric space $X$ into a metric space $Y$ and prove that $\left\{f\left(x_{n}\right)\right\}$ is a Cauchy sequence in $Y$ for every Cauchy sequence $\{x_n\}$ in $X$. |
theorem exercise_4_15 {f : β β β}
(hf : continuous f) (hof : is_open_map f) :
monotone f := | Prove that every continuous open mapping of $R^{1}$ into $R^{1}$ is monotonic. |
theorem exercise_4_21a {X : Type*} [metric_space X]
(K F : set X) (hK : is_compact K) (hF : is_closed F) (hKF : disjoint K F) :
β (Ξ΄ : β), Ξ΄ > 0 β§ β (p q : X), p β K β q β F β dist p q β₯ Ξ΄ := | Suppose $K$ and $F$ are disjoint sets in a metric space $X, K$ is compact, $F$ is closed. Prove that there exists $\delta>0$ such that $d(p, q)>\delta$ if $p \in K, q \in F$. |
theorem exercise_5_1
{f : β β β} (hf : β x y : β, | (f x - f y) | β€ (x - y) ^ 2) :
β c, f = Ξ» x, c := | Let $f$ be defined for all real $x$, and suppose that $|f(x)-f(y)| \leq (x-y)^{2}$ for all real $x$ and $y$. Prove that $f$ is constant. |
theorem exercise_5_3 {g : β β β} (hg : continuous g)
(hg' : β M : β, β x : β, | deriv g x | β€ M) :
β N, β Ξ΅ > 0, Ξ΅ < N β function.injective (Ξ» x : β, x + Ξ΅ * g x) := | Suppose $g$ is a real function on $R^{1}$, with bounded derivative (say $\left|g^{\prime}\right| \leq M$ ). Fix $\varepsilon>0$, and define $f(x)=x+\varepsilon g(x)$. Prove that $f$ is one-to-one if $\varepsilon$ is small enough. |
theorem exercise_5_5
{f : β β β}
(hfd : differentiable β f)
(hf : tendsto (deriv f) at_top (π 0)) :
tendsto (Ξ» x, f (x + 1) - f x) at_top at_top := | Suppose $f$ is defined and differentiable for every $x>0$, and $f^{\prime}(x) \rightarrow 0$ as $x \rightarrow+\infty$. Put $g(x)=f(x+1)-f(x)$. Prove that $g(x) \rightarrow 0$ as $x \rightarrow+\infty$. |
theorem exercise_5_7
{f g : β β β} {x : β}
(hf' : differentiable_at β f 0)
(hg' : differentiable_at β g 0)
(hg'_ne_0 : deriv g 0 β 0)
(f0 : f 0 = 0) (g0 : g 0 = 0) :
tendsto (Ξ» x, f x / g x) (π x) (π (deriv f x / deriv g x)) := | Suppose $f^{\prime}(x), g^{\prime}(x)$ exist, $g^{\prime}(x) \neq 0$, and $f(x)=g(x)=0$. Prove that $\lim _{t \rightarrow x} \frac{f(t)}{g(t)}=\frac{f^{\prime}(x)}{g^{\prime}(x)}.$ |
theorem exercise_5_17
{f : β β β}
(hf' : differentiable_on β f (set.Icc (-1) 1))
(hf'' : differentiable_on β (deriv f) (set.Icc 1 1))
(hf''' : differentiable_on β (deriv (deriv f)) (set.Icc 1 1))
(hf0 : f (-1) = 0)
(hf1 : f 0 = 0)
(hf2 : f 1 = 1)
(hf3 : deriv f 0 = 0) :
β x, x β set.Ioo (-1 : β) 1 β§ deriv (deriv (deriv f)) x β₯ 3 := | Suppose $f$ is a real, three times differentiable function on $[-1,1]$, such that $f(-1)=0, \quad f(0)=0, \quad f(1)=1, \quad f^{\prime}(0)=0 .$ Prove that $f^{(3)}(x) \geq 3$ for some $x \in(-1,1)$. |
theorem exercise_13_3b : Β¬ β X : Type, βs : set (set X),
(β t : set X, t β s β (set.infinite tαΆ β¨ t = β
β¨ t = β€)) β
(set.infinite (ββ s)αΆ β¨ (ββ s) = β
β¨ (ββ s) = β€) := | Show that the collection $$\mathcal{T}_\infty = \{U | X - U \text{ is infinite or empty or all of X}\}$$ does not need to be a topology on the set $X$. |
theorem exercise_13_4a2 :
β (X I : Type*) (T : I β set (set X)),
(β i, is_topology X (T i)) β§ Β¬ is_topology X (β i : I, T i) := | If $\mathcal{T}_\alpha$ is a family of topologies on $X$, show that $\bigcup \mathcal{T}_\alpha$ does not need to be a topology on $X$. |
theorem exercise_13_4b2 (X I : Type*) (T : I β set (set X)) (h : β i, is_topology X (T i)) :
β! T', is_topology X T' β§ (β i, T' β T i) β§
β T'', is_topology X T'' β (β i, T'' β T i) β T' β T'' := | Let $\mathcal{T}_\alpha$ be a family of topologies on $X$. Show that there is a unique largest topology on $X$ contained in all the collections $\mathcal{T}_\alpha$. |
theorem exercise_13_5b {X : Type*}
[t : topological_space X] (A : set (set X)) (hA : t = generate_from A) :
generate_from A = generate_from (sInter {T | is_topology X T β§ A β T}) := | Show that if $\mathcal{A}$ is a subbasis for a topology on $X$, then the topology generated by $\mathcal{A}$ equals the intersection of all topologies on $X$ that contain $\mathcal{A}$. |
theorem exercise_13_8a :
topological_space.is_topological_basis {S : set β | β a b : β, a < b β§ S = Ioo a b} := | Show that the collection $\{(a,b) \mid a < b, a \text{ and } b \text{ rational}\}$ is a basis that generates the standard topology on $\mathbb{R}$. |
theorem exercise_16_1 {X : Type*} [topological_space X]
(Y : set X)
(A : set Y) :
β U : set A, is_open U β is_open (subtype.val '' U) := | Show that if $Y$ is a subspace of $X$, and $A$ is a subset of $Y$, then the topology $A$ inherits as a subspace of $Y$ is the same as the topology it inherits as a subspace of $X$. |
theorem exercise_16_6
(S : set (set (β Γ β)))
(hS : β s, s β S β β a b c d, (rational a β§ rational b β§ rational c β§ rational d
β§ s = {x | β xβ xβ, x = (xβ, xβ) β§ a < xβ β§ xβ < b β§ c < xβ β§ xβ < d})) :
is_topological_basis S := | Show that the countable collection \[\{(a, b) \times (c, d) \mid a < b \text{ and } c < d, \text{ and } a, b, c, d \text{ are rational}\}\] is a basis for $\mathbb{R}^2$. |
theorem exercise_18_8a {X Y : Type*} [topological_space X] [topological_space Y]
[linear_order Y] [order_topology Y] {f g : X β Y}
(hf : continuous f) (hg : continuous g) :
is_closed {x | f x β€ g x} := | Let $Y$ be an ordered set in the order topology. Let $f, g: X \rightarrow Y$ be continuous. Show that the set $\{x \mid f(x) \leq g(x)\}$ is closed in $X$. |
theorem exercise_18_13
{X : Type*} [topological_space X] {Y : Type*} [topological_space Y]
[t2_space Y] {A : set X} {f : A β Y} (hf : continuous f)
(g : closure A β Y)
(g_con : continuous g) :
β (g' : closure A β Y), continuous g' β (β (x : closure A), g x = g' x) := | Let $A \subset X$; let $f: A \rightarrow Y$ be continuous; let $Y$ be Hausdorff. Show that if $f$ may be extended to a continuous function $g: \bar{A} \rightarrow Y$, then $g$ is uniquely determined by $f$. |
theorem exercise_20_2
[topological_space (β Γβ β)] [order_topology (β Γβ β)]
: metrizable_space (β Γβ β) := | Show that $\mathbb{R} \times \mathbb{R}$ in the dictionary order topology is metrizable. |
theorem exercise_21_6b
(f : β β I β β )
(h : β x n, f n x = x ^ n) :
Β¬ β fβ, tendsto_uniformly f fβ at_top := | Define $f_{n}:[0,1] \rightarrow \mathbb{R}$ by the equation $f_{n}(x)=x^{n}$. Show that the sequence $\left(f_{n}\right)$ does not converge uniformly. |
theorem exercise_22_2a {X Y : Type*} [topological_space X]
[topological_space Y] (p : X β Y) (h : continuous p) :
quotient_map p β β (f : Y β X), continuous f β§ p β f = id := | Let $p: X \rightarrow Y$ be a continuous map. Show that if there is a continuous map $f: Y \rightarrow X$ such that $p \circ f$ equals the identity map of $Y$, then $p$ is a quotient map. |
theorem exercise_22_5 {X Y : Type*} [topological_space X]
[topological_space Y] (p : X β Y) (hp : is_open_map p)
(A : set X) (hA : is_open A) : is_open_map (p β subtype.val : A β Y) := | Let $p \colon X \rightarrow Y$ be an open map. Show that if $A$ is open in $X$, then the map $q \colon A \rightarrow p(A)$ obtained by restricting $p$ is an open map. |
theorem exercise_23_3 {X : Type*} [topological_space X]
[topological_space X] {A : β β set X}
(hAn : β n, is_connected (A n))
(Aβ : set X)
(hA : is_connected Aβ)
(h : β n, Aβ β© A n β β
) :
is_connected (Aβ βͺ (β n, A n)) := | Let $\left\{A_{\alpha}\right\}$ be a collection of connected subspaces of $X$; let $A$ be a connected subset of $X$. Show that if $A \cap A_{\alpha} \neq \varnothing$ for all $\alpha$, then $A \cup\left(\bigcup A_{\alpha}\right)$ is connected. |
theorem exercise_23_6 {X : Type*}
[topological_space X] {A C : set X} (hc : is_connected C)
(hCA : C β© A β β
) (hCXA : C β© AαΆ β β
) :
C β© (frontier A) β β
:= | Let $A \subset X$. Show that if $C$ is a connected subspace of $X$ that intersects both $A$ and $X-A$, then $C$ intersects $\operatorname{Bd} A$. |
theorem exercise_23_11 {X Y : Type*} [topological_space X] [topological_space Y]
(p : X β Y) (hq : quotient_map p)
(hY : connected_space Y) (hX : β y : Y, is_connected (p β»ΒΉ' {y})) :
connected_space X := | Let $p: X \rightarrow Y$ be a quotient map. Show that if each set $p^{-1}(\{y\})$ is connected, and if $Y$ is connected, then $X$ is connected. |
theorem exercise_24_3a [topological_space I] [compact_space I]
(f : I β I) (hf : continuous f) :
β (x : I), f x = x := | Let $f \colon X \rightarrow X$ be continuous. Show that if $X = [0, 1]$, there is a point $x$ such that $f(x) = x$. (The point $x$ is called a fixed point of $f$.) |
theorem exercise_25_9 {G : Type*} [topological_space G] [group G]
[topological_group G] (C : set G) (h : C = connected_component 1) :
is_normal_subgroup C := | Let $G$ be a topological group; let $C$ be the component of $G$ containing the identity element $e$. Show that $C$ is a normal subgroup of $G$. |
theorem exercise_26_12 {X Y : Type*} [topological_space X] [topological_space Y]
(p : X β Y) (h : function.surjective p) (hc : continuous p) (hp : β y, is_compact (p β»ΒΉ' {y}))
(hY : compact_space Y) : compact_space X := | Let $p: X \rightarrow Y$ be a closed continuous surjective map such that $p^{-1}(\{y\})$ is compact, for each $y \in Y$. (Such a map is called a perfect map.) Show that if $Y$ is compact, then $X$ is compact. |
theorem exercise_28_4 {X : Type*}
[topological_space X] (hT1 : t1_space X) :
countably_compact X β limit_point_compact X := | A space $X$ is said to be countably compact if every countable open covering of $X$ contains a finite subcollection that covers $X$. Show that for a $T_1$ space $X$, countable compactness is equivalent to limit point compactness. |
theorem exercise_28_6 {X : Type*} [metric_space X]
[compact_space X] {f : X β X} (hf : isometry f) :
function.bijective f := | Let $(X, d)$ be a metric space. If $f: X \rightarrow X$ satisfies the condition $d(f(x), f(y))=d(x, y)$ for all $x, y \in X$, then $f$ is called an isometry of $X$. Show that if $f$ is an isometry and $X$ is compact, then $f$ is bijective and hence a homeomorphism. |
theorem exercise_29_4 [topological_space (β β I)] :
Β¬ locally_compact_space (β β I) := | Show that $[0, 1]^\omega$ is not locally compact in the uniform topology. |
theorem exercise_30_10
{X : β β Type*} [β i, topological_space (X i)]
(h : β i, β (s : set (X i)), countable s β§ dense s) :
β (s : set (Ξ i, X i)), countable s β§ dense s := | Show that if $X$ is a countable product of spaces having countable dense subsets, then $X$ has a countable dense subset. |
theorem exercise_31_1 {X : Type*} [topological_space X]
(hX : regular_space X) (x y : X) :
β (U V : set X), is_open U β§ is_open V β§ x β U β§ y β V β§ closure U β© closure V = β
:= | Show that if $X$ is regular, every pair of points of $X$ have neighborhoods whose closures are disjoint. |
theorem exercise_31_3 {Ξ± : Type*} [partial_order Ξ±]
[topological_space Ξ±] (h : order_topology Ξ±) : regular_space Ξ± := | Show that every order topology is regular. |
theorem exercise_32_2a
{ΞΉ : Type*} {X : ΞΉ β Type*} [β i, topological_space (X i)]
(h : β i, nonempty (X i)) (h2 : t2_space (Ξ i, X i)) :
β i, t2_space (X i) := | Show that if $\prod X_\alpha$ is Hausdorff, then so is $X_\alpha$. Assume that each $X_\alpha$ is nonempty. |
theorem exercise_32_2c
{ΞΉ : Type*} {X : ΞΉ β Type*} [β i, topological_space (X i)]
(h : β i, nonempty (X i)) (h2 : normal_space (Ξ i, X i)) :
β i, normal_space (X i) := | Show that if $\prod X_\alpha$ is normal, then so is $X_\alpha$. Assume that each $X_\alpha$ is nonempty. |
theorem exercise_33_7 {X : Type*} [topological_space X]
(hX : locally_compact_space X) (hX' : t2_space X) :
β x A, is_closed A β§ Β¬ x β A β
β (f : X β I), continuous f β§ f x = 1 β§ f '' A = {0}
:= | Show that every locally compact Hausdorff space is completely regular. |
theorem exercise_34_9
(X : Type*) [topological_space X] [compact_space X]
(X1 X2 : set X) (hX1 : is_closed X1) (hX2 : is_closed X2)
(hX : X1 βͺ X2 = univ) (hX1m : metrizable_space X1)
(hX2m : metrizable_space X2) : metrizable_space X := | Let $X$ be a compact Hausdorff space that is the union of the closed subspaces $X_1$ and $X_2$. If $X_1$ and $X_2$ are metrizable, show that $X$ is metrizable. |
theorem exercise_43_2 {X : Type*} [metric_space X]
{Y : Type*} [metric_space Y] [complete_space Y] (A : set X)
(f : X β Y) (hf : uniform_continuous_on f A) :
β! (g : X β Y), continuous_on g (closure A) β§
uniform_continuous_on g (closure A) β§ β (x : A), g x = f x := | Let $(X, d_X)$ and $(Y, d_Y)$ be metric spaces; let $Y$ be complete. Let $A \subset X$. Show that if $f \colon A \rightarrow Y$ is uniformly continuous, then $f$ can be uniquely extended to a continuous function $g \colon \bar{A} \rightarrow Y$, and $g$ is uniformly continuous. |
theorem exercise_1_2 :
(β¨-1/2, real.sqrt 3 / 2β© : β) ^ 3 = -1 := | Show that $\frac{-1 + \sqrt{3}i}{2}$ is a cube root of 1 (meaning that its cube equals 1). |
theorem exercise_1_4 {F V : Type*} [add_comm_group V] [field F]
[module F V] (v : V) (a : F): a β’ v = 0 β a = 0 β¨ v = 0 := | Prove that if $a \in \mathbf{F}$, $v \in V$, and $av = 0$, then $a = 0$ or $v = 0$. |
theorem exercise_1_7 : β U : set (β Γ β),
(U β β
) β§
(β (c : β) (u : β Γ β), u β U β c β’ u β U) β§
(β U' : submodule β (β Γ β), U β βU') := | Give an example of a nonempty subset $U$ of $\mathbf{R}^2$ such that $U$ is closed under scalar multiplication, but $U$ is not a subspace of $\mathbf{R}^2$. |
theorem exercise_1_9 {F V : Type*} [add_comm_group V] [field F]
[module F V] (U W : submodule F V):
β U' : submodule F V, (U'.carrier = βU β© βW β (U β€ W β¨ W β€ U)) := | Prove that the union of two subspaces of $V$ is a subspace of $V$ if and only if one of the subspaces is contained in the other. |
theorem exercise_3_8 {F V W : Type*} [add_comm_group V]
[add_comm_group W] [field F] [module F V] [module F W]
(L : V ββ[F] W) :
β U : submodule F V, U β L.ker = β₯ β§
linear_map.range L = range (dom_restrict L U):= | Suppose that $V$ is finite dimensional and that $T \in \mathcal{L}(V, W)$. Prove that there exists a subspace $U$ of $V$ such that $U \cap \operatorname{null} T=\{0\}$ and range $T=\{T u: u \in U\}$. |
theorem exercise_5_1 {F V : Type*} [add_comm_group V] [field F]
[module F V] {L : V ββ[F] V} {n : β} (U : fin n β submodule F V)
(hU : β i : fin n, map L (U i) = U i) :
map L (β i : fin n, U i : submodule F V) =
(β i : fin n, U i : submodule F V) := | Suppose $T \in \mathcal{L}(V)$. Prove that if $U_{1}, \ldots, U_{m}$ are subspaces of $V$ invariant under $T$, then $U_{1}+\cdots+U_{m}$ is invariant under $T$. |
theorem exercise_5_11 {F V : Type*} [add_comm_group V] [field F]
[module F V] (S T : End F V) :
(S * T).eigenvalues = (T * S).eigenvalues := | Suppose $S, T \in \mathcal{L}(V)$. Prove that $S T$ and $T S$ have the same eigenvalues. |
theorem exercise_5_13 {F V : Type*} [add_comm_group V] [field F]
[module F V] [finite_dimensional F V] {T : End F V}
(hS : β U : submodule F V, finrank F U = finrank F V - 1 β
map T U = U) : β c : F, T = c β’ id := | Suppose $T \in \mathcal{L}(V)$ is such that every subspace of $V$ with dimension $\operatorname{dim} V-1$ is invariant under $T$. Prove that $T$ is a scalar multiple of the identity operator. |
theorem exercise_5_24 {V : Type*} [add_comm_group V]
[module β V] [finite_dimensional β V] {T : End β V}
(hT : β c : β, eigenspace T c = β₯) {U : submodule β V}
(hU : map T U = U) : even (finrank U) := | Suppose $V$ is a real vector space and $T \in \mathcal{L}(V)$ has no eigenvalues. Prove that every subspace of $V$ invariant under $T$ has even dimension. |
theorem exercise_6_3 {n : β} (a b : fin n β β) :
(β i, a i * b i) ^ 2 β€ (β i : fin n, i * a i ^ 2) * (β i, b i ^ 2 / i) := | Prove that $\left(\sum_{j=1}^{n} a_{j} b_{j}\right)^{2} \leq\left(\sum_{j=1}^{n} j a_{j}{ }^{2}\right)\left(\sum_{j=1}^{n} \frac{b_{j}{ }^{2}}{j}\right)$ for all real numbers $a_{1}, \ldots, a_{n}$ and $b_{1}, \ldots, b_{n}$. |
theorem exercise_6_13 {V : Type*} [inner_product_space β V] {n : β}
{e : fin n β V} (he : orthonormal β e) (v : V) :
βvβ^2 = β i : fin n, ββͺv, e iβ«_ββ^2 β v β span β (e '' univ) := | Suppose $\left(e_{1}, \ldots, e_{m}\right)$ is an or thonormal list of vectors in $V$. Let $v \in V$. Prove that $\|v\|^{2}=\left|\left\langle v, e_{1}\right\rangle\right|^{2}+\cdots+\left|\left\langle v, e_{m}\right\rangle\right|^{2}$ if and only if $v \in \operatorname{span}\left(e_{1}, \ldots, e_{m}\right)$. |
theorem exercise_7_5 {V : Type*} [inner_product_space β V]
[finite_dimensional β V] (hV : finrank V β₯ 2) :
β U : submodule β (End β V), U.carrier β
{T | T * T.adjoint = T.adjoint * T} := | Show that if $\operatorname{dim} V \geq 2$, then the set of normal operators on $V$ is not a subspace of $\mathcal{L}(V)$. |
theorem exercise_7_9 {V : Type*} [inner_product_space β V]
[finite_dimensional β V] (T : End β V)
(hT : T * T.adjoint = T.adjoint * T) :
is_self_adjoint T β β e : T.eigenvalues, (e : β).im = 0 := | Prove that a normal operator on a complex inner-product space is self-adjoint if and only if all its eigenvalues are real. |
theorem exercise_7_11 {V : Type*} [inner_product_space β V]
[finite_dimensional β V] {T : End β V} (hT : T*T.adjoint = T.adjoint*T) :
β (S : End β V), S ^ 2 = T := | Suppose $V$ is a complex inner-product space. Prove that every normal operator on $V$ has a square root. (An operator $S \in \mathcal{L}(V)$ is called a square root of $T \in \mathcal{L}(V)$ if $S^{2}=T$.) |
theorem exercise_1_30 {n : β} :
Β¬ β a : β€, β (i : fin n), (1 : β) / (n+2) = a := | Prove that $\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n}$ is not an integer. |
theorem exercise_2_4 {a : β€} (ha : a β 0)
(f_a := | If $a$ is a nonzero integer, then for $n>m$ show that $\left(a^{2^{n}}+1, a^{2^{m}}+1\right)=1$ or 2 depending on whether $a$ is odd or even. |
theorem exercise_2_27a :
Β¬ summable (Ξ» i : {p : β€ // squarefree p}, (1 : β) / i) := | Show that $\sum^{\prime} 1 / n$, the sum being over square free integers, diverges. |
theorem exercise_3_4 : Β¬ β x y : β€, 3*x^2 + 2 = y^2 := | Show that the equation $3 x^{2}+2=y^{2}$ has no solution in integers. |
theorem exercise_3_10 {n : β} (hn0 : Β¬ n.prime) (hn1 : n β 4) :
factorial (n-1) β‘ 0 [MOD n] := | If $n$ is not a prime, show that $(n-1) ! \equiv 0(n)$, except when $n=4$. |
theorem exercise_4_4 {p t: β} (hp0 : p.prime) (hp1 : p = 4*t + 1)
(a : zmod p) :
is_primitive_root a p β is_primitive_root (-a) p := | Consider a prime $p$ of the form $4 t+1$. Show that $a$ is a primitive root modulo $p$ iff $-a$ is a primitive root modulo $p$. |
theorem exercise_4_6 {p n : β} (hp : p.prime) (hpn : p = 2^n + 1) :
is_primitive_root 3 p := | If $p=2^{n}+1$ is a Fermat prime, show that 3 is a primitive root modulo $p$. |
theorem exercise_4_11 {p : β} (hp : p.prime) (k s: β)
(s := | Prove that $1^{k}+2^{k}+\cdots+(p-1)^{k} \equiv 0(p)$ if $p-1 \nmid k$ and $-1(p)$ if $p-1 \mid k$. |
theorem exercise_5_28 {p : β} (hp : p.prime) (hp1 : p β‘ 1 [MOD 4]):
β x, x^4 β‘ 2 [MOD p] β β A B, p = A^2 + 64*B^2 := | Show that $x^{4} \equiv 2(p)$ has a solution for $p \equiv 1(4)$ iff $p$ is of the form $A^{2}+64 B^{2}$. |
theorem exercise_12_12 : is_algebraic β (sin (real.pi/12)) := | Show that $\sin (\pi / 12)$ is an algebraic number. |
theorem exercise_1_13b {f : β β β} (Ξ© : set β) (a b : Ξ©) (h : is_open Ξ©)
(hf : differentiable_on β f Ξ©) (hc : β (c : β), β z β Ξ©, (f z).im = c) :
f a = f b := | Suppose that $f$ is holomorphic in an open set $\Omega$. Prove that if $\text{Im}(f)$ is constant, then $f$ is constant. |
theorem exercise_1_19a (z : β) (hz : abs z = 1) (s : β β β)
(h : s = (Ξ» n, β i in (finset.range n), i * z ^ i)) :
Β¬ β y, tendsto s at_top (π y) := | Prove that the power series $\sum nz^n$ does not converge on any point of the unit circle. |
theorem exercise_1_19c (z : β) (hz : abs z = 1) (hz2 : z β 1) (s : β β β)
(h : s = (Ξ» n, β i in (finset.range n), i * z / i)) :
β z, tendsto s at_top (π z) := | Prove that the power series $\sum zn/n$ converges at every point of the unit circle except $z = 1$. |
theorem exercise_2_2 :
tendsto (Ξ» y, β« x in 0..y, real.sin x / x) at_top (π (real.pi / 2)) := | Show that $\int_{0}^{\infty} \frac{\sin x}{x} d x=\frac{\pi}{2}$. |
theorem exercise_2_13 {f : β β β}
(hf : β zβ : β, β (s : set β) (c : β β β), is_open s β§ zβ β s β§
β z β s, tendsto (Ξ» n, β i in finset.range n, (c i) * (z - zβ)^i) at_top (π (f zβ))
β§ β i, c i = 0) :
β (c : β β β) (n : β), f = Ξ» z, β i in finset.range n, (c i) * z ^ n := | Suppose $f$ is an analytic function defined everywhere in $\mathbb{C}$ and such that for each $z_0 \in \mathbb{C}$ at least one coefficient in the expansion $f(z) = \sum_{n=0}^\infty c_n(z - z_0)^n$ is equal to 0. Prove that $f$ is a polynomial. |
theorem exercise_3_4 (a : β) (ha : 0 < a) :
tendsto (Ξ» y, β« x in -y..y, x * real.sin x / (x ^ 2 + a ^ 2))
at_top (π (real.pi * (real.exp (-a)))) := | Show that $ \int_{-\infty}^{\infty} \frac{x \sin x}{x^2 + a^2} dx = \pi e^{-a}$ for $a > 0$. |
theorem exercise_3_14 {f : β β β} (hf : differentiable β f)
(hf_inj : function.injective f) :
β (a b : β), f = (Ξ» z, a * z + b) β§ a β 0 := | Prove that all entire functions that are also injective take the form $f(z) = az + b$, $a, b \in \mathbb{C}$ and $a \neq 0$. |
theorem exercise_5_1 (f : β β β) (hf : differentiable_on β f (ball 0 1))
(hb : bounded (set.range f)) (h0 : f β 0) (zeros : β β β) (hz : β n, f (zeros n) = 0)
(hzz : set.range zeros = {z | f z = 0 β§ z β (ball (0 : β) 1)}) :
β (z : β), tendsto (Ξ» n, (β i in finset.range n, (1 - zeros i))) at_top (π z) := | Prove that if $f$ is holomorphic in the unit disc, bounded and not identically zero, and $z_{1}, z_{2}, \ldots, z_{n}, \ldots$ are its zeros $\left(\left|z_{k}\right|<1\right)$, then $\sum_{n}\left(1-\left|z_{n}\right|\right)<\infty$. |
theorem exercise_2018_a5 (f : β β β) (hf : cont_diff β β€ f)
(hf0 : f 0 = 0) (hf1 : f 1 = 1) (hf2 : β x, f x β₯ 0) :
β (n : β) (x : β), iterated_deriv n f x = 0 := | Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be an infinitely differentiable function satisfying $f(0)=0, f(1)=1$, and $f(x) \geq 0$ for all $x \in$ $\mathbb{R}$. Show that there exist a positive integer $n$ and a real number $x$ such that $f^{(n)}(x)<0$. |
theorem exercise_2018_b4 (a : β) (x : β β β) (hx0 : x 0 = a)
(hx1 : x 1 = a)
(hxn : β n : β, n β₯ 2 β x (n+1) = 2*(x n)*(x (n-1)) - x (n-2))
(h : β n, x n = 0) :
β c, function.periodic x c := | Given a real number $a$, we define a sequence by $x_{0}=1$, $x_{1}=x_{2}=a$, and $x_{n+1}=2 x_{n} x_{n-1}-x_{n-2}$ for $n \geq 2$. Prove that if $x_{n}=0$ for some $n$, then the sequence is periodic. |
theorem exercise_2014_a5 (P : β β polynomial β€)
(hP : β n, P n = β (i : fin n), (n+1) * X ^ n) :
β (j k : β), j β k β is_coprime (P j) (P k) := | Let |
theorem exercise_2001_a5 :
β! a n : β, a > 0 β§ n > 0 β§ a^(n+1) - (a+1)^n = 2001 := | Prove that there are unique positive integers $a, n$ such that $a^{n+1}-(a+1)^n=2001$. |
theorem exercise_1999_b4 (f : β β β) (hf: cont_diff β 3 f)
(hf1 : β (n β€ 3) (x : β), iterated_deriv n f x > 0)
(hf2 : β x : β, iterated_deriv 3 f x β€ f x) :
β x : β, deriv f x < 2 * f x := | Let $f$ be a real function with a continuous third derivative such that $f(x), f^{\prime}(x), f^{\prime \prime}(x), f^{\prime \prime \prime}(x)$ are positive for all $x$. Suppose that $f^{\prime \prime \prime}(x) \leq f(x)$ for all $x$. Show that $f^{\prime}(x)<2 f(x)$ for all $x$. |
theorem exercise_1998_b6 (a b c : β€) :
β n : β€, n > 0 β§ Β¬ β m : β€, sqrt (n^3 + a*n^2 + b*n + c) = m := | Prove that, for any integers $a, b, c$, there exists a positive integer $n$ such that $\sqrt{n^3+a n^2+b n+c}$ is not an integer. |
theorem exercise_2_26 {M : Type*} [topological_space M]
(U : set M) : is_open U β β x β U, Β¬ cluster_pt x (π UαΆ) := | Prove that a set $U \subset M$ is open if and only if none of its points are limits of its complement. |
theorem exercise_2_32a (A : set β) : is_clopen A := | Show that every subset of $\mathbb{N}$ is clopen. |
theorem exercise_2_46 {M : Type*} [metric_space M]
{A B : set M} (hA : is_compact A) (hB : is_compact B)
(hAB : disjoint A B) (hAβ : A β β
) (hBβ : B β β
) :
β aβ bβ, aβ β A β§ bβ β B β§ β (a : M) (b : M),
a β A β b β B β dist aβ bβ β€ dist a b := | Assume that $A, B$ are compact, disjoint, nonempty subsets of $M$. Prove that there are $a_0 \in A$ and $b_0 \in B$ such that for all $a \in A$ and $b \in B$ we have $d(a_0, b_0) \leq d(a, b)$. |