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Theorem 2.3.1 Given infinite sequences \( \left( {{S}_{1},{S}_{2},\ldots }\right) \) and \( \left( {{T}_{1},{T}_{2},\ldots }\right) \) of sets \( {S}_{i} \) and \( {T}_{i} \) ,\n\n\[ \left\{ {H\left( {s}_{i}\right) : {s}_{i} \in {S}_{i}}\right\} = \left\{ {H\left( {t}_{i}\right) : {t}_{i} \in {T}_{i}}\right\} \text{if ... | Proof: Let \( S = \left\{ {H\left( {s}_{i}\right) : {s}_{i} \in {S}_{i}}\right\}, T = \left\{ {H\left( {t}_{i}\right) : {t}_{i} \in {T}_{i}}\right\} \) and \( Q = \left\{ {i : {S}_{i} = {T}_{i}}\right\} \). \n\nIf \( Q \in U \), take \( H\left( {s}_{i}\right) \in S \) arbitrarily, then for all \( i,{s}_{i} \in {S}_{i} ... | Yes |
Theorem 2.3.2 If for all \( i,{S}_{i} = \left\{ {s}_{i}\right\} \), then,\n\n\[ H\left( {\left\{ {s}_{1}\right\} ,\left\{ {s}_{2}\right\} ,\ldots }\right) = \left\{ {H\left( {{s}_{1},{s}_{2},\ldots }\right) }\right\} ,\text{ or }H\left( \left\{ {s}_{i}\right\} \right) = \left\{ {H\left( {s}_{i}\right) }\right\} .\n\] | Proof: Left as an exercise. | No |
Theorem 2.4.1 A constant is internal if and only if it is an element of some standard set. | Proof: The if-part is obvious. Conversely, given any internal constant \( H\left( {s}_{i}\right) = \) \( H\left( {{s}_{1},{s}_{2},\ldots }\right) \), let \( S = \left\{ {s : s = {s}_{i}}\right. \) for some \( \left. i\right\} \), then \( H\left( {s}_{i}\right) \in {}^{ * }S \) . | Yes |
Theorem 2.4.2 Let \( S \) be a classical set such that \( {}^{ * }s = s \) for each \( s \in S \) . Then \( {}^{ * }S = S \) if and only if \( S \) is finite. Otherwise \( S \subset {}^{ * }S \) . | Proof: Since \( * s = H\left( s\right) = s \) if \( s \in S, * S = \left\{ {H\left( S\right) = H\left( {s}_{i}\right) : {s}_{i} \in S}\right\} \supseteq S \) . If \( S \) is finite then any \( H\left( {s}_{i}\right) = s \) for some \( s \in S \), hence \( {}^{ * }S \subseteq S \), and if \( S \) is infinite then there ... | Yes |
Corollary 2.4.1 Let \( S \) be a classical set, such that each of its elements is a classical set of which each element is equal to its \( * \) -transform. Then \( {}^{ * }S = S \) if and only if \( S \) is finite and all its elements are finite. A similar result holds if \( S \) is of any level (see Section 1.13 for a... | Proof: Left as an exercise. | No |
Corollary 2.4.3 Let \( g \) be a function from \( W \) to \( X \), let \( f \) be a function from \( X \) onto \( Y \), and assume that \( X \) is finite, that \( {}^{ * }x = x \) for all \( x \in X \) and that \( {}^{ * }y = y \) for all \( y \in Y \) . Then, with \( {w}_{i}, w \in W \) ,\n\n\[ \n{}^{ * }\left( {f \ci... | Proof: It follows that \( {}^{ * }X = X,{}^{ * }Y = Y \) and that \( {}^{ * }f = f \) . Since \( {}^{ * }\left( {f \circ g}\right) = {}^{ * }f \circ {}^{ * }g \), the first equality follows immediately, and since \( {}^{ * }g\left( {{}^{ * }w}\right) { = }^{ * }\left( {g\left( w\right) }\right) = g\left( w\right) \) al... | Yes |
Theorem 2.5.1 If \( S \) is a classical infinite set of numbers, then \( S \) is external. | Proof: First notice that if \( T \) is any set of classical numbers, then \( {}^{ * }T \supseteq T \), because if \( t \in T \) then \( * t = H\left( t\right) = t \) . This implies that if \( T \subseteq S \), then \( S \cap * T = T \), for if \( s \in S \cap {}^{ * }T \) then \( s = {}^{ * }s \in {}^{ * }T \), so that... | Yes |
Theorem 2.7.1 (Loš’ theorem.)\n\nLet any classical statement,\n\n\[ R\left( {X,{X}^{\prime },{X}^{\prime \prime },\ldots ;s,{s}^{\prime },{s}^{\prime \prime },\ldots }\right) ,\]\n\nwith a finite number of constants or free variables \( X,{X}^{\prime },{X}^{\prime \prime },\ldots, s,{s}^{\prime },{s}^{\prime \prime },\... | In other words, (regarding for the time being each constant as a free variable) given any classical statement, to each of its free variables \( q \) (so that here \( q \in \) \( \left. \left\{ {X,{X}^{\prime },{X}^{\prime \prime },\ldots, s,{s}^{\prime },{s}^{\prime \prime },\ldots }\right\} \right) \) add the index \(... | Yes |
Corollary 2.7.1 (The internal definition principle.) Let any statement be given, say \( P\left( {x, X, s}\right) \), where \( X \) is some set and \( x \in X \) makes sense. Let \( X \) and \( s \) be internal, then so is the set \[ T = \{ x \in X : P\left( {x, X, s}\right) \} . \] | Proof: Since \( X \) is internal, also \( x \) is internal, so for suitable \( {X}_{i} \) and \( {s}_{i} \) , \[ T = \left\{ {H\left( {x}_{i}\right) \in H\left( {X}_{i}\right) : P\left( {H\left( {x}_{i}\right), H\left( {X}_{i}\right), H\left( {s}_{i}\right) }\right) }\right\} . \] Let, \[ {T}_{i} = \left\{ {{x}_{i} \in... | Yes |
Theorem 2.8.1 (Transfer, first formulation.)\n\nLet \( R\left( {X,{X}^{\prime },{X}^{\prime \prime },\ldots ;s,{s}^{\prime },{s}^{\prime \prime },\ldots }\right) \) be as before. Then,\n\n\[ R\left( {X,{X}^{\prime },{X}^{\prime \prime },\ldots ;s,{s}^{\prime },{s}^{\prime \prime },\ldots }\right) \equiv \]\n\n\[ R\left... | Proof: In Loš’ theorem take \( {X}_{i} = X,{X}_{i}^{\prime } = {X}^{\prime },{X}_{i}^{\prime \prime } = {X}^{\prime \prime },\ldots ,{s}_{i} = s,{s}_{i}^{\prime } = {s}^{\prime } \) , \( {s}_{i}^{\prime \prime } = {s}^{\prime \prime },\ldots \), for all \( i \), then,\n\n\[ {}^{ * }\left\lbrack {R\left( {X,{X}^{\prime ... | Yes |
Given any internal statement, replacing everything, including every bound variable, by its standard version is equivalent to replacing everything except every bound variable by its standard version. | \[ {\exists }^{ * }x \in {}^{ * }X : P\left( {{}^{ * }x,{}^{ * }s}\right) \equiv \exists x \in {}^{ * }X : P\left( {x,{}^{ * }s}\right) ,\] and \[ {\forall }^{ * }x \in {}^{ * }X : P\left( {{}^{ * }x,{}^{ * }s}\right) \equiv \forall x \in {}^{ * }X : P\left( {x,{}^{ * }s}\right) ,\] where \( H\left( {x}_{i}\right) \) h... | Yes |
Theorem 2.8.3 (The standard definition principle.)\n\nLet \( {}^{ * }X \) be a standard set, \( x \in {}^{ * }X \) make sense, \( {}^{ * }s \) be standard, and \( P\left( {x,{}^{ * }X,{}^{ * }s}\right) \) be any statement. Then also,\n\n\[ \left\{ {x \in {}^{ * }X : P\left( {x,{}^{ * }X,{}^{ * }s}\right) }\right\} \]\n... | Proof: The proof is a simplification of the proof of the internal definition principle, and is left as an exercise. | No |
Theorem 2.10.1 Infinitely large numbers and nonzero infinitesimals exist! | Proof: For example,\n\n\[ H\left( {1,2,3,\ldots }\right) \sim \infty, H\left( {-1, - 2, - 3,\ldots }\right) \sim - \infty ,\]\n\n\[ H\left( {1,1/2,1/3,\ldots }\right) \sim 0, H\left( {-1, - 1/2, - 1/3,\ldots }\right) \sim 0.\]\n\nClearly \( x = H\left( {x}_{i}\right) = H\left( {+1, - 1/2, + 1/3, - 1/4,\ldots }\right) \... | Yes |
Theorem 2.10.2 \( \varepsilon \sim 0 \) if and only if \( 1/\varepsilon \sim + \infty \) or \( 1/\varepsilon \sim - \infty \), hence \( s \) is appreciable if and only if \( 1/s \) is appreciable. | Let \( \varepsilon \sim 0,{\varepsilon }^{\prime } \sim 0, s \) and \( {s}^{\prime } \) be appreciable, and \( \omega \sim \infty ,{\omega }^{\prime } \sim \infty \) . Then,\n\n\[ \varepsilon + {\varepsilon }^{\prime } \simeq 0,\varepsilon - {\varepsilon }^{\prime } \simeq 0,\varepsilon \cdot {\varepsilon }^{\prime } \... | No |
Theorem 2.10.4 In \( {}^{ * }\mathbb{R} \) the set of all positive hyperlarge numbers is external. | Proof: Left as an exercise. Hint: use lower bounds. | No |
Theorem 2.11.1 (Overflow or overspill.) Let \( S \) be an internal subset of \( {}^{ * }T \), where \( T \) is either \( \mathbb{N} \) or \( \mathbb{Z} \), or \( \mathbb{Q} \), or \( \mathbb{R} \), such that \( \forall m \in \mathbb{N} : \exists s\left( m\right) \in S : s\left( m\right) \geq m \), i.e. such that from a... | Proof: If \( \forall b \in {}^{ * }T : \exists s\left( b\right) \in S : s\left( b\right) > b \), then take \( b \sim \infty \), which implies that \( s\left( b\right) \sim \infty \) . If this is not true, then \( \exists b \in {}^{ * }T : \forall s \in S : s \leq b \), so that by Loš’ theorem, \( H\left( {\exists {b}_{... | Yes |
Theorem 2.11.2 (Underflow or underspill.) Let \( S \) be an internal subset of \( {}^{ * }T \), with \( T \) as before, such that \( \forall \omega \in {}^{ * }\mathbb{N},\omega \sim \) \( \infty : \exists s\left( \omega \right) \in S : s\left( \omega \right) < \omega \land s\left( \omega \right) \sim \infty \), i.e. s... | Proof: Let \( {S}_{1} = \{ s \in S : s \geq 1\} \) . Clearly, \( {S}_{1} \) is not empty, so that by the classical greatest lower bound theorem, \[ \exists \beta \in {}^{ * }\mathbb{N} : \left\lbrack {\forall s \in {S}_{1} : s \geq \beta }\right\rbrack \land \left\lbrack {\exists {s}^{\prime } \in {S}_{1} : {s}^{\prime... | Yes |
Theorem 2.11.3 ('Inverse' overflow.)\n\nLet \( S \) be an internal subset of ’ \( \mathbb{Q} \) or \( {}^{ * }\mathbb{R} \), such that \( \forall m \in \mathbb{N} : \exists s\left( m\right) \in S \) : \( \left| {s\left( m\right) }\right| \leq 1/m \), i.e. such that from a classical point of view \( S \) contains arbitr... | Proof: It is no restriction to assume that \( 0 \notin S \) . Apply overflow to \( {S}^{\prime } = \{ t : 1/t \in \) \( S\} \) and use the fact that \( {S}^{\prime } \) is internal if (and only if) \( S \) is internal. | No |
Theorem 2.11.4 ('Inverse' underflow.) Let \( S \) be an internal subset of Q or \( {}^{ * }\mathbb{R} \), such that \( \forall \varepsilon ,\varepsilon \sim 0,\varepsilon > 0 : \exists s \in S : s \geq \varepsilon \) , then \( \exists s \in S : s \) is not an infinitesimal. | Proof: Similar to the preceding proof. | No |
Theorem 2.12.1 If \( n \in \mathbb{Q} \), then either \( n \in \mathbb{N} \) or \( n \) is hyperlarge. | Proof: If \( n = H\left( {n}_{i}\right) \) is not hyperlarge, then \( n \leq m \) for some \( m \in \mathbb{N} \), so that, \( \left\{ {i : 0 < {n}_{i} \leq m}\right\} \in U \), but then \( \left\{ {i : {n}_{i} = {m}^{\prime }}\right\} \in U \) for precisely one \( {m}^{\prime } \in \mathbb{N} \) , \( {m}^{\prime } \le... | Yes |
Corollary 2.12.1 If \( x \in {}^{ * }\mathbb{N} \) and \( x \sim \infty \), then there exists a nondecreasing infinite sequence \( \left( {x}_{i}\right) \) tending to infinity such that \( x = H\left( {x}_{i}\right) \) . | Example: Let \( x = H\left( {y}_{i}\right) \), where \( {y}_{i} = 1 \) if \( i = {2j} + 1 \) for some \( j,{y}_{i} = 2 \) if \( i = {4j} + 2 \) for some \( j,{y}_{i} = 3 \) for \( i = {8j} + 4 \) for some \( j \), etc. Hence for each \( n \) the number of \( {y}_{i} = n \) is infinitely large, and \( \left( {y}_{i}\rig... | Yes |
Corollary 2.12.2 * \( \mathbb{N} \) is already generated by all nondecreasing infinite sequences that tend to infinity. | Proof: The proof uses a variation of Cantor’s diagonal method. If * \( \mathbb{N} \) were countable, let \( s\left( n\right) = H\left( {{s}_{i}\left( n\right) }\right) = H\left( {{s}_{1}\left( n\right) ,{s}_{2}\left( n\right) ,\ldots }\right) \) be its \( n \) -th element, \( n \in \mathbb{N} \) . Let,\n\n\[ \n{t}_{1} ... | Yes |
Theorem 2.12.3 Let \( f \) be an internal function from \( {}^{ * }\mathbb{N} \) to \( \{ 1,2\} \), such that both 1 and 2 are assumed somewhere, i.e. such that \( f \) is onto. Moreover, let \( f\left( n\right) = 2 \) for all \( n \geq b \) for some \( b \in {}^{ * }\mathbb{N} \) . Then there is a \( \beta < b,\beta \... | Proof: The set \( \left\{ {x \in {}^{ * }\mathbb{N} : f\left( x\right) = 1}\right\} \) is internal and is bounded above by \( b \), hence by the least upper bound theorem in its internal form it has a least upper bound \( \beta \), which must be a maximum. | No |
Theorem 2.13.1 (Standard part theorem.) If \( x \in {}^{ * }\mathbb{R} \) then either \( \left| x\right| \sim \infty \), or \( x = r + \varepsilon, r \in \mathbb{R},\varepsilon \simeq 0 \) for unique \( r \) and \( \varepsilon \) . | Proof: Let \( x \) not be hyperlarge. Then the uniqueness of \( r \) and \( \varepsilon \) is easily shown, for let \( r + \varepsilon = {r}^{\prime } + {\varepsilon }^{\prime }, r,{r}^{\prime } \in \mathbb{R},\varepsilon ,{\varepsilon }^{\prime } \simeq 0 \), then \( r - {r}^{\prime } \simeq 0 \), but \( r - {r}^{\pri... | Yes |
Theorem 2.13.2 Given any \( r \in \mathbb{R} \), there exists an \( x \in \mathbb{Q} \) such that \( \operatorname{st}\left( x\right) = r \) . | Proof: If \( r \in \mathbb{R} \), then there exists a sequence \( \left( {r}_{n}\right) ,{r}_{n} \in \mathbb{Q} \), such that \( {r}_{n} \) tends to \( r \) if \( n \) tends to infinity, hence for all \( m \in \mathbb{N},\left| {{r}_{n} - r}\right| < 1/m \) if \( n \) is large enough. This implies that,\n\n\[ \forall m... | Yes |
Theorem 2.13.3 Let \( a, b \in \mathbb{R}, a < b \) . If \( x \in * \left\lbrack {a, b}\right\rbrack \), then \( \operatorname{st}\left( x\right) \in \left\lbrack {a, b}\right\rbrack \), but if the interval in \( R \) is not closed, this is sometimes not true. | Proof: Since \( x \) is limited, \( \operatorname{st}\left( x\right) \) is well defined. If \( \operatorname{st}\left( x\right) \notin \left\lbrack {a, b}\right\rbrack \), then either \( \operatorname{st}\left( x\right) = \) \( a - \delta \) or \( \operatorname{st}\left( x\right) = b + \delta \) for some \( \delta \in ... | Yes |
Theorem 3.1.1 (The least upper bound theorem.)\n\nLet \( S \) be a nonempty subset of \( \mathbb{R} \) that is bounded above by some (classical) real number. Then \( S \) has a least upper bound in \( \mathbb{R} \) . | Proof: Taking any \( c \in S \), instead of \( S \) we may consider \( \{ s : s \in S, s \geq c\} \) , that is to say we may assume that \( s \geq c \) for all \( s \in S \) . Then \( c, b \in \mathbb{R}, c < b \) , exist such that \( \forall s \in S : c \leq s \leq b \), so that, by transfer, \( \forall s{ \in }^{ * }... | Yes |
Theorem 3.2.1 (Simplified definition of the continuity of real-valued real functions.) \( f : \mathbb{R} \rightarrow \mathbb{R} \) is continuous at \( c \in \mathbb{R} \) if and only if, | \[ \forall \delta \in {}^{ * }\mathbb{R},\delta \simeq 0 : {}^{ * }f\left( {c + \delta }\right) - {}^{ * }f\left( c\right) \simeq 0. \] | Yes |
Theorem 3.2.2 (Simplified definition of uniform continuity.) \( f : S \rightarrow \mathbb{R}, S \subseteq \mathbb{R} \) is uniformly continuous in \( S \) if and only if,\n\n\[ \forall x, y \in {}^{ * }S, x - y \simeq 0 : {}^{ * }f\left( x\right) - {}^{ * }f\left( y\right) \simeq 0. \] | Proof: Recall that \( f : S \rightarrow \mathbb{R}, S \subseteq \mathbb{R} \) is uniformly continuous in \( S \) if,\n\n\[ \forall \varepsilon \in \mathbb{R},\varepsilon > 0,\exists \delta \in \mathbb{R},\delta > 0 : \forall x, y \in S,\left| {x - y}\right| < \delta : \left| {f\left( x\right) - f\left( y\right) }\right... | Yes |
Theorem 3.2.3 If \( f \) is continuous at each \( x \in \left\lbrack {a, b}\right\rbrack, a, b \in \mathbb{R}, a < b \), then \( f \) is uniformly continuous in \( \left\lbrack {a, b}\right\rbrack \) . | Simplified proof: Let \( x, y \) be in \( {}^{ * }\left\lbrack {a, b}\right\rbrack \), then by Theorem 2.13.3, \( \operatorname{st}\left( x\right) \in \left\lbrack {a, b}\right\rbrack \) and \( \operatorname{st}\left( y\right) \in \left\lbrack {a, b}\right\rbrack \) . Let \( x - y \simeq 0 \) . Since \( y - \operatorna... | Yes |
Theorem 3.2.4 (Simplified limit definition.)\n\nLet \( f : \mathbb{R} \rightarrow \mathbb{R} \), then \( \mathop{\lim }\limits_{{x \rightarrow c}}f\left( x\right) = k, c, k \in \mathbb{R} \) if and only if,\n\n\[ \forall \delta \in {}^{ * }\mathbb{R},\delta \sim 0 : {}^{ * }f\left( {c + \delta }\right) - k \simeq 0, \]... | Proof: By definition, the limit exists if and only if,\n\n\[ \forall \varepsilon \in \mathbb{R},\varepsilon > 0 : \exists \delta \in \mathbb{R},\delta > 0 : \forall x \in \mathbb{R},0 < \mid x - c \mid < \delta : \left| {f\left( x\right) - k}\right| < \varepsilon ,\]\n\nor, by transfer,\n\n\[ \forall \varepsilon \in {}... | Yes |
Theorem 3.2.5 (Another simplified limit definition.) Let \( f : \mathbb{N} \rightarrow \mathbb{R} \) and \( k \in \mathbb{R} \), then \( \mathop{\lim }\limits_{{x \rightarrow \infty }}f\left( n\right) = k, k \in \mathbb{R} \), if and only if, \[ \forall n \in {}^{ * }\mathbb{N}, n \sim \infty : {}^{ * }f\left( n\right)... | Proof: By definition, the limit exists if and only if, \[ \forall \varepsilon \in \mathbb{R},\varepsilon > 0 : \exists {n}^{\prime } \in \mathbb{N} : \forall n \in \mathbb{N}, n > {n}^{\prime } : \left| {f\left( n\right) - k}\right| < \varepsilon , \] or, by transfer, \[ \forall \varepsilon \in {}^{ * }\mathbb{R},\vare... | Yes |
Theorem 3.2.6 If \( f \) is a nondecreasing infinite sequence, that is bounded above, then \( f\left( n\right) \) has a finite limit for \( n \) tending to \( \infty \) . | Classical proof: The set \( \{ f\left( n\right) : n \in \mathbb{N}\} \) is bounded above, hence in \( \mathbb{R} \) has a least upper bound \( \beta \), so that \( f\left( n\right) \leq \beta \) for all \( n \in \mathbb{N} \), and for each \( m \in \mathbb{N} \) , \( f\left( {n}^{\prime }\right) > \beta - 1/m \) for so... | Yes |
Theorem 3.2.7 (The intermediate value theorem.) If \( a, b \in \mathbb{R}, a < b \), and \( f\left( a\right) < 0, f\left( b\right) > 0 \), then \( f\left( c\right) = 0 \) for some \( c, a < c < b \) . | Simplified proof: See Section 1.4. | No |
Theorem 3.2.8 (The extreme value theorem.) Let \( f : \left\lbrack {a, b}\right\rbrack \rightarrow \mathbb{R}, a, b \in \mathbb{R}, a < b \), and let \( f \) be continuous at each point of \( \left\lbrack {a, b}\right\rbrack \) . Then \( f\left( x\right) \leq f\left( c\right) \) for some \( c \in \left\lbrack {a, b}\ri... | Simplified proof: Let \( \omega \in {}^{ * }\mathbb{N},\omega \sim \infty \), be arbitrary, and divide \( {}^{ * }\left\lbrack {a, b}\right\rbrack \) in \( \omega \) equal subintervals of length \( \delta = \left( {b - a}\right) /\omega \) . Let \( n \in {}^{ * }\mathbb{N} \) be such that \( {}^{ * }f\left( {a + {n\del... | Yes |
Theorem 3.2.9 (The composite function theorem.)\n\nLet \( g\left( w\right) \) be defined for \( w \) in a neighborhood of \( c \in \mathbb{R} \), and let \( f\left( x\right) \) be defined for \( x \) in a neighborhood of \( g\left( c\right) \) . Then \( f \circ g \) is continuous at \( c \) if \( g \) is continuous at ... | Simplified proof: Let \( \delta \simeq 0 \), then \( {}^{ * }g\left( {c + \delta }\right) - g\left( c\right) \simeq 0 \), hence it follows that \( {}^{ * }f\left( {{}^{ * }g\left( {c + \delta }\right) }\right) - f\left( {g\left( c\right) }\right) \simeq 0. \n\n## 3.3 Continuity and limits for internal functions | No |
Theorem 3.3.1 (Continuity of internal functions.) The internal \( f : {}^{ * }\mathbb{R} \rightarrow {}^{ * }\mathbb{R} \) is * continuous at \( c \in {}^{ * }\mathbb{R} \) if and only if in the classical definition \( \mathbb{R} \) is replaced by \( {}^{ * }\mathbb{R} \), i.e. if, \[ \forall \varepsilon \in {}^{ * }\m... | Proof: Letting \( f = H\left( {f}_{i}\right) \) and \( c = H\left( {c}_{i}\right), f \) is *continuous at \( c \) if and only if, \[ H\left\lbrack {\forall {\varepsilon }_{i} \in \mathbb{R},{\varepsilon }_{i} > 0 : \exists {\delta }_{i} \in \mathbb{R},{\delta }_{i} > 0 : }\right. \] \[ \forall {x}_{i} \in \mathbb{R},\l... | Yes |
Theorem 3.3.2 The internal function \( f : {}^{ * }\mathbb{R} \rightarrow {}^{ * }\mathbb{R} \) is \( S \) -continuous at \( c \in {}^{ * }\mathbb{R} \) if and only if,\n\n\[ \forall \varepsilon \in \mathbb{R},\varepsilon < 0 : \exists \delta \in \mathbb{R},\delta > 0 : \forall x \in {}^{ * }\mathbb{R},\left| {x - c}\r... | Proof: The if part. Let \( \varepsilon \in \mathbb{R},\varepsilon > 0 \) and \( \delta \in {}^{ * }\mathbb{R},\delta \simeq 0 \) be given arbitrarily. Then there is a \( {\delta }^{\prime } \in \mathbb{R},{\delta }^{\prime } > 0 \) such that,\n\n\[ \forall x \in {}^{ * }\mathbb{R},\left| {x - c}\right| < {\delta }^{\pr... | Yes |
Theorem 3.3.3 The internal \( f : {}^{ * }\mathbb{R} \rightarrow {}^{ * }\mathbb{R} \) tends to the \( S \) -limit \( k \in {}^{ * }\mathbb{R} \) for \( x \in {}^{ * }\mathbb{R} \) tending to \( c \in {}^{ * }\mathbb{R} \), if and only if,\n\n\[ \forall \varepsilon \in \mathbb{R},\varepsilon > 0 : \exists \delta \in \m... | Proof: Left as an exercise. | No |
Theorem 3.3.4 The internal \( f : {}^{ * }\mathbb{N} \rightarrow {}^{ * }\mathbb{R} \) tends to the \( S \) -limit \( k \in {}^{ * }\mathbb{R} \) for \( n \in {}^{ * }\mathbb{N} \) tending to infinity, if and only if,\n\n\[ \forall \varepsilon \in \mathbb{R},\varepsilon > 0 : \exists {n}^{\prime } \in \mathbb{N} : \for... | Proof: Left as an exercise. | No |
Theorem 3.3.5 Let \( f, c \) and \( k \) be as before, and let \( k \) and \( c \) be finite. If \( f\left( x\right) \) tends to \( k \) for \( x \in {}^{ * }\mathbb{R} \) tending to \( c \), then, \[ \mathop{\lim }\limits_{{x \rightarrow {st}\left( c\right) }}{st}\left( {f\left( x\right) }\right) = {st}\left( k\right)... | Proof: In the first case \[ \exists {\delta }_{1} \in \mathbb{R},{\delta }_{1} > 0 : \forall x \in {}^{ * }\mathbb{R},0 < \left| {x - c}\right| < {\delta }_{1} : \mid f\left( x\right) - k \mid < 1, \] so that \( \forall x \in \mathbb{R},0 < \left| {x - \operatorname{st}\left( c\right) }\right| < {\delta }_{1}/2 : \left... | Yes |
Theorem 3.4.1 (Nonstandard characterization of Cauchy sequence.) \( s\left( n\right) \) is a classical Cauchy sequence if and only if, \[ \forall n, p \in {}^{ * }\mathbb{N}, n, p \sim \infty : {}^{ * }s\left( n\right) - {}^{ * }s\left( p\right) \simeq 0. \] | Proof: By definition, \[ \forall m \in \mathbb{N} : \exists k \in \mathbb{N} : \forall n, p \in \mathbb{N}, n, p > k : \mid s\left( n\right) - s\left( p\right) \mid < 1/m, \] and by transfer, fixing \( m \in \mathbb{N} \) and \( k \in \mathbb{N} \) , \[ \forall n, p \in \mathbb{N}, n, p > k : \mid s\left( n\right) - s\... | Yes |
Theorem 3.4.2 (Nonstandard characterization of bounded set.) Let \( L \) be the set of all limited elements of \( {}^{ * }\mathbb{R} \) . Then \( S \subseteq \mathbb{R} \) is bounded if and only if \( {}^{ * }S \subseteq L \) . | Proof: \( S \) is bounded if,\n\n\[ \exists m \in \mathbb{N} : \forall s \in S : \left| s\right| \leq m \]\n\nhence, by transfer, if,\n\n\[ \exists m \in \mathbb{N} : \forall s \in {}^{ * }S : \left| s\right| \leq m \]\n\nso that \( {}^{ * }S \subseteq L \) .\n\nConversely, if \( S \) is not bounded then,\n\n\[ \forall... | Yes |
Theorem 3.4.3 (Nonstandard characterization of open set.)\n\nLet \( S \subseteq \mathbb{R} \) and let \( h\left( S\right) = \{ t \in * \mathbb{R} : t \simeq s \) for some \( s \in S\} \) . Then \( S \) is open if and only if,\n\n\[ h\left( S\right) \subseteq {}^{ * }S\text{.} \] | Proof: \( S \) is open if,\n\n\[ \forall s \in S : \exists m \in \mathbb{N} : \forall t \in \mathbb{R},\left| {t - s}\right| < 1/m : t \in S, \]\n\nhence, by transfer,\n\n\[ \forall s \in S : \exists m \in \mathbb{N} : \forall t \in {}^{ * }\mathbb{R},\left| {t - s}\right| < 1/m : t \in {}^{ * }S, \]\n\nso that, restri... | Yes |
Theorem 3.4.4 (Nonstandard characterization of closed set.)\n\n\( S \subseteq \mathbb{R} \) is closed if and only if,\n\n\[ h\left( {S}^{c}\right) \subseteq {}^{ * }\left( {S}^{c}\right) = {\left( {}^{ * }S\right) }^{c}. \] | Proof: Follows directly from the previous theorem. | No |
Theorem 3.4.5 (Nonstandard characterization of interior point.) Let \( s \in \mathbb{R} \) and let \( h\left( s\right) = \{ t \in \mathbb{R} : t \simeq s\} \) . Then \( s \) is an interior point of \( S \subseteq \mathbb{R} \) if and only if \( h\left( s\right) \subseteq {}^{ * }S \) . | Proof: Since \( s \) is an interior point of \( S \) if, \[ \exists m \in \mathbb{N} : \forall t \in \mathbb{R},\left| {t - s}\right| < 1/m : t \in S, \] the proof is a simplified version of that of Theorem 3.4.3. The details are left as an exercise. | No |
Theorem 3.4.6 (Nonstandard characterization of boundary point.)\n\n\\( s \\in \\mathbb{R} \\) is a boundary point of \\( S \\subseteq \\mathbb{R} \\) if and only if both \\( h\\left( s\\right) \\cap {}^{ * }S \\) and \\( h\\left( s\\right) \\cap {\\left( {}^{ * }S\\right) }^{c} \\) are nonempty. | Proof: If \\( s \\) is a boundary point of \\( S \\) then,\n\n\\[ \n\\forall m \\in \\mathbb{N} : \\left\\lbrack {\\exists t \\in \\mathbb{R},\\left| {t - s}\\right| < 1/m : t \\in S}\\right\\rbrack \\land \\left\\lbrack {\\exists t \\in \\mathbb{R},\\left| {t - s}\\right| < 1/m : t \\notin S}\\right\\rbrack \n\\]\n\nh... | Yes |
Theorem 3.4.7 (Nonstandard characterizations of accumulation point and closure.)\n\n\( s \in \mathbb{R} \) is an accumulation point (or limit point) of \( S \subseteq \mathbb{R} \) if and only if,\n\n\[ \exists t \in {}^{ * }S, t \neq s : t \sim s. \]\n\nLet \( {clS} \) be the closure of \( S \) . Then \( s \in {clS} \... | Proof: If \( s \) is an accumulation point of \( S \) then,\n\n\[ \forall m \in \mathbb{N} : \exists t \in S, t \neq s : \mid t - s \mid < 1/m, \]\n\nhence, by transfer,\n\n\[ \forall m \in {}^{ * }\mathbf{N} : \exists t \in {}^{ * }S, t \neq s : \mid t - s \mid < 1/m, \]\n\nso that, taking \( m \) hyperlarge, \( \exis... | Yes |
Theorem 3.5.1 Let a function \( f \) be monotonically increasing (or decreasing) and be continuous in \( \left\lbrack {a, b}\right\rbrack, a, b \in \mathbb{R}, a < b \) . Then,\n\n1) range \( \left( f\right) \), the range of \( f \), is a finite closed interval,\n\n1) \( \;f \) has an inverse,\n\n1) \( \;{f}^{-1} \) to... | Proof: Only the case where \( f \) is increasing is considered.\n\n1) As \( a \leq x \leq b \) implies that \( f\left( a\right) \leq f\left( x\right) \leq f\left( b\right) \), range \( \left( f\right) \subseteq \left\lbrack {f\left( a\right), f\left( b\right) }\right\rbrack \) . And if \( f\left( a\right) \leq w \leq f... | Yes |
1) If \( f : \mathbb{R} \rightarrow \mathbb{R} \) is differentiable at \( c \in \mathbb{R} \), then \( f \) is continuous at \( c \) . | 1) Follows immediately from \( {}^{ * }f\left( {c + \delta }\right) - f\left( c\right) - {f}^{\prime }\left( c\right) \cdot \delta = {\tau \delta } \), so that, \n\n\[ \n{}^{ * }f\left( {c + \delta }\right) - f\left( c\right) \simeq 0\text{ if }\delta \simeq 0. \n\] | No |
Theorem 3.6.2 (The critical point theorem.)\n\nLet \( X \) be an interval of \( \mathbb{R}, c \in X, f : X \rightarrow \mathbb{R} \) be continuous at each \( x \in X \), and \( c \) be a maximum or a minimum of \( f \) over \( X \) . Then \( c \) is an endpoint of \( X \), or \( {f}^{\prime }\left( c\right) \) is undef... | Proof: Let \( c \) be a maximum of \( f \) over \( X \) . If \( c \) is not an endpoint of \( X \) and if \( {f}^{\prime }\left( c\right) \) exists, then,\n\n\[ \forall \delta \sim 0 : {f}^{\prime }\left( c\right) = \operatorname{st}\left\lbrack \frac{*f\left( {c + \delta }\right) - f\left( c\right) }{\delta }\right\rb... | Yes |
Theorem 3.6.3 (Rolle's theorem.)\n\nLet \( f \) be continuous in \( \left\lbrack {a, b}\right\rbrack, a, b \in \mathbb{R}, a < b \), and be differentiable in \( \left( {a, b}\right) \) . Moreover, let \( f\left( a\right) = f\left( b\right) = 0 \) . Then \( {f}^{\prime }\left( c\right) = 0 \) for some \( c \in \left( {a... | Proof: The proof is classical in nature, and relies on Theorems 3.2.8 and 3.6.2. | No |
Corollary 3.6.1 (Mean value theorem.)\n\nLet \( f \) be as in the previous theorem, except \( f\left( a\right) = f\left( b\right) = 0 \) need not be true. Then,\n\n\[ \n{f}^{\prime }\left( c\right) = \frac{f\left( b\right) - f\left( a\right) }{b - a}\text{ for some }c \in \left( {a, b}\right) .\n\] | Proof: The proof follows by applying Rolle’s theorem to \( g \), defined by,\n\n\[ \ng\left( x\right) = f\left( x\right) - f\left( a\right) \cdot \frac{f\left( b\right) - f\left( a\right) }{b - a} \cdot \left( {x - a}\right) .\n\] | Yes |
Corollary 3.6.2 (Generalized mean value theorem.) Let \( f \) and \( g \) both be continuous in \( \left\lbrack {a, b}\right\rbrack \), and be differentiable in \( \left( {a, b}\right) \) . In addition let \( {g}^{\prime }\left( x\right) \neq 0 \) for all \( x \in \left( {a, b}\right) \) . Then, | Proof: The proof follows by applying Rolle’s theorem to \( h \), defined by,\n\n\[ h\left( x\right) = f\left( x\right) \left( {g\left( b\right) - g\left( a\right) }\right) - g\left( x\right) \left( {f\left( b\right) - f\left( a\right) }\right) . \] | Yes |
Corollary 3.6.3 (Taylor's theorem.) Let \( f \) and \( {f}^{\prime } \) both be continuous in \( \left\lbrack {a, b}\right\rbrack \), and let \( {f}^{\prime \prime } \) exist in \( \left( {a, b}\right) \), then, \[ f\left( b\right) = f\left( a\right) + {f}^{\prime }\left( a\right) \left( {b - a}\right) /1! + {f}^{\prim... | Proof: The proof is based on the previous corollary. | No |
Theorem 3.6.4 (L'Hopital's theorem).\n\nLet both \( f \) and \( g \) be differentiable, and \( {g}^{\prime }\left( x\right) \neq 0 \) in a neighborhood \( \left( {a, b}\right) \) of\n\n\( c \in \mathbb{R} \) with the possible exception of the point \( c \) itself. Also assume that \( \mathop{\lim }\limits_{{x \rightarr... | Proof: Only the case as stated will be proved.\n\nWithout loss of generality it may be assumed that \( f\left( x\right) \) and \( g\left( x\right) \) are defined at \( x = c \) and that \( f\left( x\right) = g\left( x\right) = 0 \), so that \( f \) and \( g \) are continuous at \( c \) . It follows that the generalized... | No |
Theorem 3.7.1 If \( a < b < c \), then,\n\n\[ \n{\int }_{a}^{c}f\left( x\right) {dx} = {\int }_{a}^{b}f\left( x\right) {dx} + {\int }_{b}^{c}f\left( x\right) {dx}. \n\] | Proof: Follows from the fact that the standard part of a sum is equal to the sum of the standard parts. | No |
Theorem 3.7.2 \( F \) defined by \( F\left( x\right) = {\int }_{a}^{x}f\left( t\right) {dt}, a \leq x \leq b, a < b \), is continuous. Moreover, the derivative of \( F\left( x\right) \) exists and is equal to \( f\left( x\right), a < x < b \) . Conversely, if \( G \) is such that \( {G}^{\prime }\left( x\right) = f\lef... | Proof: The continuity of \( F \) follows from the fact that, if \( \delta \simeq 0 \) ,\n\n\[ {\int }_{a}^{x + \delta }f\left( t\right) {dt} - {\int }_{a}^{x}f\left( t\right) {dt} = {\int }_{x}^{x + \delta }f\left( t\right) {dt} \]\n\nno matter whether \( \delta \) is nonnegative or negative, in which case the right-ha... | Yes |
Theorem 3.7.3 (Substitution rule.)\n\nLet \( {F}^{\prime }\left( x\right) = f\left( x\right) \) for \( x \in \left\lbrack {a, b}\right\rbrack \), and let \( x = g\left( w\right) \) for \( w \in \left\lbrack {\alpha ,\beta }\right\rbrack \) such that \( g \) maps \( \left\lbrack {\alpha ,\beta }\right\rbrack \) onto \( ... | Proof: From Theorem 3.6.1 it follows that,\n\n\[ \n{\left( F\left( g\left( w\right) \right) \right) }^{\prime } = {F}^{\prime }\left( {g\left( w\right) }\right) {g}^{\prime }\left( w\right) = f\left( {g\left( w\right) }\right) {g}^{\prime }\left( w\right) , \n\]\n\nhence,\n\n\[ \n{\int }_{a}^{b}f\left( x\right) {dx} = ... | Yes |
Theorem 3.7.4 (Integration by parts.)\n\nIf both \( f \) and \( g \) are continuously differentiable in \( \left\lbrack {a, b}\right\rbrack \), then,\n\n\[ \n{\int }_{a}^{b}f\left( x\right) {g}^{\prime }\left( x\right) {dx} = f\left( b\right) g\left( b\right) - f\left( a\right) g\left( a\right) - {\int }_{a}^{b}{f}^{\p... | Proof: Since \( {\left( f\left( x\right) g\left( x\right) \right) }^{\prime } = f\left( x\right) {g}^{\prime }\left( x\right) + {f}^{\prime }\left( x\right) g\left( x\right) \) it follows that,\n\n\[ \nJ = {\int }_{a}^{b}{\left( f\left( x\right) g\left( x\right) \right) }^{\prime }{dx}\n\]\n\nexists and is equal to the... | Yes |
Theorem 4.1.1 If \( f : X \rightarrow {}^{ * }\mathbb{R} \) is an internal function such that \( \forall x \in X \) : \( f\left( x\right) \simeq 0 \), then \( \mathop{\sup }\limits_{{x \in X}}\left| {f\left( x\right) }\right| \simeq 0 \) . | Proof 1: Since \( f\left( x\right) \) is bounded over \( X \), the supremum exists. Denote it by \( \beta \) . Then \( \forall \delta > 0 : \exists x \in X : \mid f\left( x\right) \mid \geq \beta - \delta \), that is \( \beta \leq \delta + \mid f\left( x\right) \mid \) . Let \( \delta \sim 0 \), then it follows that \(... | Yes |
Corollary 4.1.1 Let \( \left\lbrack {a, b}\right\rbrack \) be some interval of \( {}^{ * }\mathbb{R} \), such that \( b - a \) is limited, and let \( f \) and \( g \) both be Riemann integrable functions over \( \left\lbrack {a, b}\right\rbrack \) such that \( f\left( x\right) \simeq g\left( x\right) \) for all \( x \i... | Proof: Let \( \beta = \mathop{\sup }\limits_{{a \leq x \leq b}}\left| {f\left( x\right) - g\left( x\right) }\right| \) . According to the theorem \( \beta \simeq 0 \), hence, \[ 0 \leq \left| {{\int }_{a}^{b}f\left( x\right) {dx} - {\int }_{a}^{b}g\left( x\right) {dx}}\right| \leq {\int }_{a}^{b}\left| {f\left( x\right... | Yes |
Theorem 4.1.4 (Fehrele's principle.) No halo is a galaxy, hence no galaxy is a halo. | Proof: (Van den Berg.) Assume to the contrary that some halo \( H \) is equal to some galaxy \( G \) . Let \( H = { \cap }_{n \in \mathbf{N}}S\left( n\right) \) and \( G = { \cup }_{n \in \mathbf{N}}T\left( n\right) \), with \( S\left( n\right) \) and \( T\left( n\right) \) internal, \( \left( {S\left( n\right) }\right... | Yes |
Let \( \left( {s\left( n\right), n \in {}^{ * }\mathbb{N}}\right), s\left( n\right) \in {}^{ * }\mathbb{R} \), be an internal sequence, such that \( s\left( n\right) \simeq 0 \) for all \( n \in \mathbb{N} \) . Then, \[ \exists \omega \in {}^{ * }\mathbb{N},\omega \sim \infty : \forall k \in {}^{ * }\mathbb{N}, k \leq ... | Proof 1: (Van den Berg.) Let \( H = \left\{ {n \in {}^{ * }\mathbb{N} : \left\lbrack {\forall k \leq n : s\left( k\right) \simeq 0}\right\rbrack }\right\} \) and \( G = \mathbb{N} \) . Then \( G \) is a galaxy and \( H \supseteq \mathbb{N} = G \) . If \( H \) is external, then \( G \subset H \) (by Fehrele’s principle)... | Yes |
Corollary 4.1.3 (Dominated approximation.) Let \( f, g \) and \( h \) be functions from \( {}^{ * }\mathbb{R} \) to \( {}^{ * }\mathbb{R} \), Riemann integrable over \( \left( {-\infty , + \infty }\right) \) . Let \( f \) and \( g \) be internal but \( h \) be standard. Assume that \( f\left( x\right) \simeq g\left( x\... | Proof: Given any \( n \in \mathbb{N} \), let \( \beta = \mathop{\sup }\limits_{{\left| x\right| \leq n}}\left| {f\left( x\right) - g\left( x\right) }\right| \), then \( \beta \simeq 0 \), as follows from Theorem 4.1.1, and Corollary 4.1.1 implies that, \[ \forall n \in \mathbb{N} : {\int }_{-n}^{+n}f\left( x\right) {dx... | Yes |
Theorem 4.2.1 (Saturation.)\n\nLet the infinite sequence \( \left( {S\left( n\right), n \in \mathbb{N}}\right) \) of internal sets \( S\left( n\right) \) have the finite intersection property, then the intersection of all of them is nonempty, i.e. \( { \cap }_{n \in \mathbf{N}}S\left( n\right) \neq \) 0. | Proof 1: (Not using permanence.)\n\nLet \( S\left( n\right) = H\left( {{S}_{i}\left( n\right) }\right) \), where \( H \) is the \( H \) - operator of the basic theory. For all \( n \in \mathbb{N} \), let,\n\n\[ T\left( n\right) = { \cap }_{k = 1}^{n}S\left( k\right) ,\text{ and }{T}_{i}\left( n\right) = { \cap }_{k = 1... | Yes |
Corollary 4.2.1 Let \( A \) be a given internal set, and \( \left( {S\left( n\right) }\right) \) an infinite sequence of internal subsets of \( A \) . If for all \( n \in \mathbb{N},{ \cup }_{k = 1}^{n}S\left( k\right) \neq A \), then \( { \cup }_{k \in \mathbf{N}}S\left( k\right) \neq A \) . Hence if the union of any ... | Proof: The proof follows from the fact that \( {\left( \cup S\left( n\right) \right) }^{c} = \cap {S}^{c}\left( n\right) \), where \( c \) denotes complementation with respect to \( A \) . | No |
Corollary 4.2.2 Given an infinite sequence \( \left( {S\left( n\right) }\right) \) of internal sets, then \( S = \) \( { \cup }_{k \in \mathbf{N}}S\left( k\right) \) is internal if and only if there exists an \( n \in \mathbf{N} \) such that \( S = { \cup }_{k = 1}^{n}S\left( k\right) \) . | Proof: The if-part follows immediately. Conversely, if \( S \) is internal, then so are all \( T\left( n\right) = S - S\left( n\right) \), and \( { \cap }_{k \in \mathbf{N}}T\left( k\right) = \varnothing \), hence there must exist an \( n \in \mathbb{N} \) such that \( { \cap }_{k = 1}^{n}T\left( k\right) = \varnothing... | Yes |
Given an infinite sequence \( \left( {S\left( n\right) }\right) \) of internals sets, then \( S = \) \( { \cap }_{k \in \mathbf{N}}S\left( k\right) \) is internal if and only if there exists an \( n \in \mathbb{N} \) such that \( S = { \cap }_{k = 1}^{n}S\left( k\right) \) . | Proof: By complementation from Corollary 4.2.1. | No |
Example 1.1.1 : The notion of equality satisfies these axioms of an equivalence relation. So \( a \sim b \) iff \( a = b \) is an equivalence relation. | The main point, however, is that an equivalence relation is a more flexible notion than equality, and yet captures many of the important aspects of equality. | No |
Example 1.1.4 : At the other extreme from equality we could say that every element of the set \( X \) is equivalent to every other element. This would be the coarsest possible equivalence relation on the set. | Example 1.1.4' : If we take the coarsest possible equivalence relation then there is only one equivalence class, namely the full set \( X \) itself. | Yes |
Suppose \( G \) is a group. Suppose \( X \) is any set. Consider the set of all functions from \( X \) to \( G \) :\n\n\[ \mathcal{F} = \{ f : f\text{ is a function from }X \rightarrow G\} \]\n\nIf we want to stress the role of \( X \) and/or \( G \) we write \( \mathcal{F}\left\lbrack {X \rightarrow G}\right\rbrack \)... | The main step to show this is simply giving a definition of the group multiplication and the inversion operation. The product \( {\mathbf{m}}_{\mathcal{F}}\left( {{f}_{1},{f}_{2}}\right) \) of two functions \( {f}_{1},{f}_{2} \in \mathcal{F} \) must be another function in \( \mathcal{F} \) . We define this function by ... | Yes |
Example 3.2 The sign homomorphism. | \[ \epsilon : {S}_{n} \rightarrow {\mathbb{Z}}_{2} \] where we identify \( {\mathbb{Z}}_{2} \) as the multiplicative group \( \{ \pm 1\} \) of square roots of 1 . The rule is: \( \epsilon : \sigma \rightarrow + 1 \) if \( \sigma \) is a product of an even number of transpositions. \( \epsilon : \sigma \rightarrow - 1 \... | Yes |
Theorem 6.1.1 (Lagrange) If \( H \) is a subgroup of a finite group \( G \) then the order of \( H \) divides the order of \( G \) : | Proof : If \( G \) is finite \( G = { \coprod }_{1}^{m}{g}_{i}H \) for some set of \( {g}_{i} \), leading to distinct cosets. Now note that the order of any coset is the order of \( H \), because the invertible action of left-multiplication by \( g \) sets up a 1-1 correspondence between the elements of \( H \) and tho... | Yes |
Consider \( G = U\left( N\right) \) . Then conjugacy within \( U\left( N\right) \) is the same as unitary equivalence: \( {u}_{1},{u}_{2} \in U\left( N\right) \) are conjugate if there is a \( g \in U\left( N\right) \) with \( {u}_{2} = g{u}_{1}{g}^{ \dagger } = g{u}_{1}{g}^{-1} \) . | An important theorem, the Spectral theorem, proved by induction on \( N \) in section 17 of the Linear Algebra User’s Manuel, implies that if \( u \in U\left( N\right) \) there is a \( g \in U\left( N\right) \) with \( {gu}{g}^{-1} = \operatorname{Diag}\left\{ {{z}_{1},\ldots ,{z}_{N}}\right\} \) where \( \left| {z}_{i... | No |
Now consider \( G = {GL}\left( {n,\mathbb{C}}\right) \) . We must stress that not all matrices are diagonalizable, so that the full description of conjugacy classes is more complicated. For any matrix \( A \in {M}_{n}\left( \mathbb{C}\right) \) we can define its characteristic polynomial\n\n\[ \n{p}_{A}\left( x\right) ... | If \( r \) is a root of this polynomial then the matrix \( {r1} - A \) has zero determinant, so it has a nontrivial kernel (see Chapter 2) and therefore there is an eigenvector \( v \) of \( A \) with eigenvalue \( r \) :\n\n\[ \n{Av} = {rv} \n\]\n\nConversely, any eigenvalue of \( A \) must be a root of the polynomial... | No |
Theorem 6.2.1. If \( N \subset G \) is a normal subgroup then the set of left cosets \( G/N = \) \( \{ {gN} \mid g \in G\} \) has a natural group structure with group multiplication defined by:\n\n\[ \n\left( {{g}_{1}N}\right) \cdot \left( {{g}_{2}N}\right) \mathrel{\text{:=}} \left( {{g}_{1} \cdot {g}_{2}}\right) N \n... | Proof- left as an important exercise - see below. | No |
Example 6.2.1 Cyclic Groups Since \( \mathbb{Z} \) is Abelian \( n\mathbb{Z} \subset \mathbb{Z} \) is normal, and the quotient group is \( \mathbb{Z}/n\mathbb{Z} \) . This is isomorphic to the cyclic group we have previously denoted as \( {\mu }_{n} \) or \( {\mathbb{Z}}_{n} \) . So \( \bar{r} \) is the equivalence cla... | \[ \bar{r} = r + n\mathbb{Z} \] \( \left( {6.35}\right) \) \[ \bar{r} + \bar{s} = \left( {r + s}\right) + n\mathbb{Z} \] \( \left( {6.36}\right) \) | Yes |
Consider the subgroup \( H \subset G \) of all integral linear combinations of \( {f}_{i} \) :\n\n\[ H \mathrel{\text{:=}} \left\{ {\mathop{\sum }\limits_{{i = 1}}^{d}{n}_{i}{f}_{i} \mid {n}_{i} \in \mathbb{Z}}\right\} \] | If \( \det A \neq 0 \) then in fact \( G/H \) is a finite group. One way to see this easily is to consider \( G \) as a subgroup of \( {\mathbb{Q}}^{d} \), so that we can write\n\n\[ {e}_{i} = {A}_{ij}^{-1}{f}_{j} \]\n\nwith \( {A}^{-1} \in {GL}\left( {d,\mathbb{Q}}\right) \) . Recall that \( {A}^{-1} = {\left( \det A\... | Yes |
Recall that \( {SL}\left( {n,\kappa }\right) \subset {GL}\left( {n,\kappa }\right) ,{SO}\left( {n,\kappa }\right) \subset O\left( {n,\kappa }\right) \), and \( {SU}\left( n\right) \subset \) \( U\left( n\right) \) are all subgroups defined by the condition \( \det A = 1 \) on a matrix. Note that, since \( \det \left( {... | \[ {GL}\left( {n,\kappa }\right) /{SL}\left( {n,\kappa }\right) \cong {\kappa }^{ * } \] \[ O\left( {n,\mathbb{R}}\right) /{SO}\left( {n,\kappa }\right) \cong {\mathbb{Z}}_{2} \] (6.54) \[ U\left( n\right) /{SU}\left( n\right) \cong U\left( 1\right) \] Lines 1 and 3 follow since every element in \( {GL}\left( {n,\kappa... | Yes |
Let \( G \) be a topological group. Let \( {G}_{0} \) be the (path-) connected component of the identity element \( {1}_{G} \in G \). We claim that \( {G}_{0} \) is a normal subgroup. | If \( {g}_{0} \in {G}_{0} \) there is a continuous path of group elements \( \gamma : \left\lbrack {0,1}\right\rbrack \rightarrow G \) with \( \gamma \left( 0\right) = {1}_{G} \) and \( \gamma \left( 1\right) = {g}_{0} \). Then if \( g \in G \) is any other group element \( {g\gamma }\left( t\right) {g}^{-1} \) is a co... | Yes |
The center of \( U\left( N\right) \) consists of matrices proportional to the unit matrix. | See the exercise below. | No |
Example 2 Similarly, we can write an indecomposable representation of the connected component of the identity of the \( 1 + 1 \) dimensional Lorentz group. If \( B\left( \eta \right) \in S{O}_{0}\left( {1,1}\right) \) is the boost of rapidity \( \eta \), with \( \eta \in \mathbb{R} \), then: | \[ T\left( {B\left( \eta \right) }\right) = \left( \begin{array}{ll} 1 & \eta \\ 0 & 1 \end{array}\right) \] | Yes |
Example 2 As a somewhat trivial special case of the above consider \( \mathbb{Z} \) acting on \( \mathbb{R} : x \rightarrow \) \( - x \) therefore acts on \( \operatorname{Fun}\left( \mathbb{R}\right) \) . Decompose into irreps: | \( \psi = \) function on \( \mathbb{R} \)\n\n\[ \n{\psi }^{1}\left( x\right) = \frac{1}{2}\left( {\psi \left( x\right) + \psi \left( {-x}\right) }\right) \n\]\n\n\( \left( {10.225}\right) \)\n\n\[ \n{\psi }^{2}\left( x\right) = \frac{1}{2}\left( {\psi \left( x\right) - \psi \left( {-x}\right) }\right) \n\]\n\ntransform... | Yes |
Consider the \( 3 \times 3 \) rep generated by the \( \natural \) action of the permutation group \( {S}_{3} \) on \( {\mathbb{R}}^{3} \) . We’ll compute the characters by choosing one representative from each conjugacy class: | \[ 1 \rightarrow \left( \begin{array}{lll} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right) \;\left( {12}\right) \rightarrow \left( \begin{array}{lll} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{array}\right) \;\left( {132}\right) \rightarrow \left( \begin{array}{lll} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{array}\... | Yes |
Extensions of \( {\mathbb{Z}}_{2} \) by \( {\mathbb{Z}}_{2} \) . WLOG we can take \( f\left( {1,1}\right) = f\left( {1,\sigma }\right) = f\left( {\sigma ,1}\right) = 1 \) . Then we have two choices: \( f\left( {\sigma ,\sigma }\right) = 1 \) or \( f\left( {\sigma ,\sigma }\right) = \sigma \) . Each of these choices sat... | Indeed \( \sigma \) is an involution and also \( \sigma \) is not a perfect square, so by our discussion above a cocycle with \( f\left( {\sigma ,\sigma }\right) = \sigma \) cannot be gauged to the trivial cocycle. In other words \( {H}^{2}\left( {{\mathbb{Z}}_{2},{\mathbb{Z}}_{2}}\right) = {\mathbb{Z}}_{2} \) . For th... | Yes |
Example 1. The simplest example is where we have a symmetry group \( G = {\mathbb{Z}}_{2} \) interpreted as time reversal. It will be convenient to denote \( {M}_{2} = \{ 1,\bar{T}\} \), with \( {\bar{T}}^{2} = 1 \) . Of course, \( {M}_{2} \cong {\mathbb{Z}}_{2} \) . In quantum mechanics the representation of \( \bar{T... | \[ {H}^{2 + \omega }\left( {{\mathbb{Z}}_{2}, U\left( 1\right) }\right) = {\mathbb{Z}}_{2} \] (14.590) To prove this we look at the twisted cocycle identity. Exactly the same arguments as in Remark 5 of section 14.3 show that we can choose a gauge with \( f\left( {g,1}\right) = f\left( {1, g}\right) = 1 \) for all \( g... | Yes |
Consider time evolution in quantum mechanics with a time-dependent Hamiltonian. There is no sense to time evolution \( U\left( t\right) \) . Rather one must speak of unitary evolution \( U\left( {{t}_{1},{t}_{2}}\right) \) such that \( U\left( {{t}_{1},{t}_{2}}\right) U\left( {{t}_{2},{t}_{3}}\right) = U\left( {{t}_{1}... | Given a solution of the Schrodinger equation \( \Psi \left( t\right) \) we may consider the state vectors \( \Psi \left( t\right) \) as objects and \( U\left( {{t}_{1},{t}_{2}}\right) \) as morphisms. In this way a solution of the Schrodinger equation defines a groupoid. | Yes |
In the theory of string theory orbifolds and orientifolds spacetime must be considered to be a groupoid. Suppose we have a right action of \( G \) on a set \( X \), so we have a map\n\n\[ \Phi : X \times G \rightarrow X \]\n\n(16.37)\n\nsuch that\n\n\[ \Phi \left( {\Phi \left( {x,{g}_{1}}\right) ,{g}_{2}}\right) = \Phi... | We should think of a morphism as an arrow, labeled by \( g \), connecting the point \( x \) to the point \( x \cdot g \) . The target and source maps are:\n\n\[ {p}_{0}\left( \left( {x, g}\right) \right) \mathrel{\text{:=}} x \cdot g\;{p}_{1}\left( \left( {x, g}\right) \right) \mathrel{\text{:=}} x \]\n\n(16.41)\n\nThe... | Yes |
Lemma 1.1.1 A minimal path \( x : \left\lbrack {{t}_{0},{t}_{1}}\right\rbrack \rightarrow {\mathbb{R}}^{n} \) is a solution of the Euler-Lagrange equations\n\n\[ \frac{d}{dt}\frac{\partial L}{\partial v} = \frac{\partial L}{\partial x} \] | Proof: If \( x \) is minimal, all directional derivatives of \( I \) at \( x \) must vanish. Hence\n\n\[ 0 = {\left. \frac{d}{d\varepsilon }\right| }_{\varepsilon = 0}I\left( {x + {\varepsilon \xi }}\right) \]\n\n\[ = {\int }_{{t}_{0}}^{{t}_{1}}\left( {\mathop{\sum }\limits_{{j = 1}}^{n}\frac{\partial L}{\partial {x}_{... | Yes |
Lemma 1.1.8 A curve \( z : \left\lbrack {{t}_{0},{t}_{1}}\right\rbrack \rightarrow {\mathbb{R}}^{2n} \) is a critical point of \( {\mathcal{A}}_{H} \) (with respect to variations with fixed endpoints \( x\left( {t}_{0}\right) = {x}_{0} \) and \( \left. {x\left( {t}_{1}\right) = {x}_{1}}\right) \) if and only if it sati... | Proof: Let \( {z}_{s} = \left( {{x}_{s},{y}_{s}}\right) : \left\lbrack {{t}_{0},{t}_{1}}\right\rbrack \rightarrow {\mathbb{R}}^{2n} \) be a 1-parameter family of curves satisfying \( {x}_{s}\left( {t}_{0}\right) = {x}_{0} \) and \( {x}_{s}\left( {t}_{1}\right) = {x}_{1} \) for all \( s \) and \( {z}_{0} = z \) . Denote... | Yes |
Lemma 1.1.9 Let \( \psi : {\mathbb{R}}^{2n} \rightarrow {\mathbb{R}}^{2n} \) be a symplectomorphism and let \( H : {\mathbb{R}}^{2n} \rightarrow \mathbb{R} \) be a smooth Hamiltonian function. Suppose that \( \zeta \left( t\right) \) is a solution of the Hamiltonian differential equation\n\n\[ \dot{\zeta } = {X}_{H \ci... | Proof: An easy calculation shows that, for any diffeomorphism \( \psi \) of \( {\mathbb{R}}^{2n} \) and any function \( H \), the gradient \( \nabla H \) transforms by\n\n\[ \nabla \left( {H \circ \psi }\right) \left( p\right) = {d\psi }{\left( p\right) }^{\mathrm{T}}\nabla H\left( {\psi \left( p\right) }\right) .\n\nH... | Yes |
Lemma 1.1.10 Wherever defined, \( {\phi }_{H}^{t,{t}_{0}} \) is a symplectomorphism. | Proof: Let \( {z}_{0} \in {\mathbb{R}}^{2n} \) and define \( z\left( t\right) \mathrel{\text{:=}} {\phi }_{H}^{t,{t}_{0}}\left( {z}_{0}\right) \) and\n\n\[ \Phi \left( t\right) \mathrel{\text{:=}} d{\phi }_{H}^{t,{t}_{0}}\left( {z}_{0}\right) \in {\mathbb{R}}^{{2n} \times {2n}}. \]\n\nThen for every \( {\zeta }_{0} \in... | Yes |
Lemma 1.1.11 If \( \Phi \) and \( \Psi \) are symplectic matrices then so are \( {\Phi \Psi },{\Psi }^{-1},{\Psi }^{\mathrm{T}} \) . | Proof: Multiply the identity \( {\Psi }^{\mathrm{T}}{J}_{0}\Psi = {J}_{0} \) by \( {\Psi }^{-1} \) from the right and by \( {\left( {\Psi }^{\mathrm{T}}\right) }^{-1} \) from the left to obtain \( {J}_{0} = {\left( {\Psi }^{\mathrm{T}}\right) }^{-1}{J}_{0}{\Psi }^{-1} \) . So \( {\Psi }^{-1} \) is a symplectic matrix. ... | Yes |
Lemma 1.1.12 \( \operatorname{Sp}\left( {2n}\right) \) is a Lie group with Lie algebra | Proof: Denote by \( \mathfrak{{so}}\left( {2n}\right) \subset {\mathbb{R}}^{{2n} \times {2n}} \) the linear subspace of skew-symmetric \( {2n} \times {2n} \) matrices and define the map \( f : {\mathbb{R}}^{{2n} \times {2n}} \rightarrow \mathfrak{{so}}\left( {2n}\right) \) by\n\n\[ f\left( \Psi \right) \mathrel{\text{:... | Yes |
Lemma 1.1.15 Every symplectic matrix has determinant 1. | Proof: Let \( \Psi \in \operatorname{Sp}\left( {2n}\right) \) . Then \( {\Psi }^{T}{J}_{0}\Psi = {J}_{0} \) and hence it follows from equation (1.1.21) that \( {\omega }_{0}\left( {{\Psi \zeta },\Psi {\zeta }^{\prime }}\right) = {\omega }_{0}\left( {\zeta ,{\zeta }^{\prime }}\right) \) for all \( \zeta ,{\zeta }^{\prim... | Yes |
Example 1.1.17 (Hamiltonian flow on the sphere) The level sets of the function\n\n\[ \nH = \frac{1}{2}\mathop{\sum }\limits_{{j = 1}}^{n}\left( {{x}_{j}^{2} + {y}_{j}^{2}}\right) \n\]\n\nare spheres centred at the origin. The symplectic gradient is the vector field \( {X}_{H}\left( z\right) = - {J}_{0}z \) and the Hami... | The easiest way to understand the corresponding flow is to identify \( {\mathbb{R}}^{2n} \) with complex \( n \) -space \( {\mathbb{C}}^{n} \) via \( z = \left( {{z}_{1},\ldots ,{z}_{n}}\right) = \left( {{x}_{1} + i{y}_{1},\ldots ,{x}_{n} + i{y}_{n}}\right) \) . Then \( {J}_{0} \) acts by multiplication with \( i \) an... | Yes |
Lemma 1.1.18 The Poisson bracket defines a Lie algebra structure on the vector space \( {C}^{\infty }\left( {\mathbb{R}}^{2n}\right) \), i.e. it is skew-symmetric and satisfies the Jacobi identity\n\n\[ \{ \{ F, G\}, H\} + \{ \{ G, H\}, F\} + \{ \{ H, F\}, G\} = 0. \] | Proof of Lemma 1.1.18: Denote by \( {d}^{2}F\left( z\right) = d\nabla F\left( z\right) \) the Hessian of \( F \) at \( z \) , understood as the symmetric \( {2n} \times {2n} \) -matrix of second partial derivatives of \( F \) . Then\n\n\[ \nabla \{ F, G\} = - \nabla \left( {{\left( \nabla F\right) }^{T}{J}_{0}\nabla G}... | Yes |
Lemma 1.1.19 A diffeomorphism \( \psi : {\mathbb{R}}^{2n} \rightarrow {\mathbb{R}}^{2n} \) is a symplectomorphism if and only if\n\n\[ \{ F, G\} \circ \psi = \{ F \circ \psi, G \circ \psi \} \]\n\nfor all \( F, G \in {C}^{\infty }\left( {\mathbb{R}}^{2n}\right) \) . | Proof: Let \( \psi : {\mathbb{R}}^{2n} \rightarrow {\mathbb{R}}^{2n} \) be a diffeomorphism. Then, by (1.1.23),\n\n\[ \{ F, G\} \circ \psi = - {\left( \nabla F \circ \psi \right) }^{T}{J}_{0}\left( {\nabla G \circ \psi }\right) ,\]\n\n\[ \{ F \circ \psi ,\mathrm{G} \circ \psi \} = - \nabla {\left( F \circ \psi \right) ... | Yes |
Example 1.1.23 (Geodesic flow) Geodesics on a Riemannian manifold are defined as paths which locally minimize length. We now show that they are given by a Hamiltonian system of equations. | More precisely, consider the variational problem with the Lagrangian\n\n\[ L\left( {x, v}\right) = \frac{1}{2}{v}^{\mathrm{T}}g\left( x\right) v = \frac{1}{2}\mathop{\sum }\limits_{{i, j = 1}}^{n}{g}_{ij}\left( x\right) {v}_{i}{v}_{j}, \]\n\nwhere the matrix \( {g}_{ij}\left( x\right) = {g}_{ji}\left( x\right) \) is po... | Yes |
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