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Type | Label | Description |
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Statement | ||
Theorem | climinf 27101* | A bounded monotonic non increasing sequence converges to the infimum of its range. (Contributed by Glauco Siliprandi, 29-Jun-2017.) |
Theorem | climsuselem1 27102* | The subsequence index has the expected properties: it belongs to the same upper integers as the original index, and it is always larger or equal than the original index. (Contributed by Glauco Siliprandi, 29-Jun-2017.) |
Theorem | climsuse 27103* | A subsequence of a converging sequence , converges to the same limit. is the strictly increasing and it is used to index the subsequence (Contributed by Glauco Siliprandi, 29-Jun-2017.) |
Theorem | climrecf 27104* | A version of climrec 27098 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 29-Jun-2017.) |
Theorem | climneg 27105* | Complex limit of the negative of a sequence. (Contributed by Glauco Siliprandi, 29-Jun-2017.) |
Theorem | climinff 27106* | A version of climinf 27101 using bound-variable hypotheses instead of distinct variable conditions (Contributed by Glauco Siliprandi, 29-Jun-2017.) |
Theorem | climdivf 27107* | Limit of the ratio of two converging sequences. (Contributed by Glauco Siliprandi, 29-Jun-2017.) |
Theorem | climreeq 27108 | If is a real function, then converges to with respect to the standard topology on the reals if and only if it converges to with respect to the standard topology on complex numbers. In the theorem, is defined to be convergence w.r.t. the standard topology on the reals and then represents the statement " converges to , with respect to the standard topology on the reals". Notice that there is no need for the hypothesis that is a real number. (Contributed by Glauco Siliprandi, 2-Jul-2017.) |
Theorem | dvsinexp 27109* | The derivative of sin^N . (Contributed by Glauco Siliprandi, 29-Jun-2017.) |
Theorem | dvcosre 27110 | The real derivative of the cosine (Contributed by Glauco Siliprandi, 29-Jun-2017.) |
Theorem | ioovolcl 27111 | An open real interval has finite volume. (Contributed by Glauco Siliprandi, 29-Jun-2017.) |
Theorem | volioo 27112 | The measure of an open interval. (Contributed by Glauco Siliprandi, 29-Jun-2017.) |
Theorem | itgsin0pilem1 27113* | Calculation of the integral for sine on the (0,π) interval (Contributed by Glauco Siliprandi, 29-Jun-2017.) |
Theorem | ibliccsinexp 27114* | sin^n on a closed interval is integrable. (Contributed by Glauco Siliprandi, 29-Jun-2017.) |
Theorem | itgsin0pi 27115 | Calculation of the integral for sine on the (0,π) interval (Contributed by Glauco Siliprandi, 29-Jun-2017.) |
Theorem | iblioosinexp 27116* | sin^n on an open integral is integrable (Contributed by Glauco Siliprandi, 29-Jun-2017.) |
Theorem | itgsinexplem1 27117* | Integration by parts is applied to integrate sin^(N+1) (Contributed by Glauco Siliprandi, 29-Jun-2017.) |
Theorem | itgsinexp 27118* |
A recursive formula for the integral of sin^N on the interval (0,π) .
(Contributed by Glauco Siliprandi, 29-Jun-2017.) |
Theorem | stoweidlem1 27119 | Lemma for stoweid 27181. This lemma is used by Lemma 1 in [BrosowskiDeutsh] p. 90; the key step uses Bernoulli's inequality bernneq 11194. (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
Theorem | stoweidlem2 27120* | lemma for stoweid 27181: here we prove that the subalgebra of continuous functions, which contains constant functions, is closed under scaling. (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
Theorem | stoweidlem3 27121* | Lemma for stoweid 27181: if is positive and all terms of a finite product are larger than , then the finite product is larger than A^M. (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
Theorem | stoweidlem4 27122* | Lemma for stoweid 27181: a class variable replaces a set variable, for constant functions. (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
Theorem | stoweidlem5 27123* | There exists a δ as in the proof of Lemma 1 in [BrosowskiDeutsh] p. 90: 0 < δ < 1 , p >= δ on . Here is used to represent δ in the paper and to represent in the paper. (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
Theorem | stoweidlem6 27124* | Lemma for stoweid 27181: two class variables replace two set variables, for multiplication of two functions. (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
Theorem | stoweidlem7 27125* | This lemma is used to prove that q_{n} as in the proof of Lemma 1 in [BrosowskiDeutsh] p. 91, (at the top of page 91), is such that q_{n} < ε on , and q_{n} > 1 - ε on . Here it is proven that, for large enough, 1-(k*δ/2)^n > 1 - ε , and 1/(k*δ)^n < ε. The variable is used to represent (k*δ) in the paper, and is used to represent (k*δ/2). (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
Theorem | stoweidlem8 27126* | Lemma for stoweid 27181: two class variables replace two set variables, for the sum of two functions. (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
Theorem | stoweidlem9 27127* | Lemma for stoweid 27181: here the Stone Weierstrass theorem is proven for the trivial case, T is the empty set. (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
Theorem | stoweidlem10 27128 | Lemma for stoweid 27181. This lemma is used by Lemma 1 in [BrosowskiDeutsh] p. 90, this lemma is an application of Bernoulli's inequality. (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
Theorem | stoweidlem11 27129* | This lemma is used to prove that there is a function as in the proof of [BrosowskiDeutsh] p. 92, (at the top of page 92): this lemma proves that g(t) < ( j + 1 / 3 ) * ε. Here is used to represent ε in the paper. (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
Theorem | stoweidlem12 27130* | Lemma for stoweid 27181. This Lemma is used by other three Lemmas. (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
Theorem | stoweidlem13 27131 | Lemma for stoweid 27181. This lemma is used to prove the statement abs( f(t) - g(t) ) < 2 epsilon , in [BrosowskiDeutsh] p. 92, the last step of the proof. (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
Theorem | stoweidlem14 27132* | There exists a as in the proof of Lemma 1 in [BrosowskiDeutsh] p. 90: is an integer and 1 < k * δ < 2. is used to represent δ in the paper. (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
Theorem | stoweidlem15 27133* | This lemma is used to prove the existence of a function as in Lemma 1 from [BrosowskiDeutsh] p. 90: is in the subalgebra, such that 0 ≤ p ≤ 1, p_{(t}_0) = 0, and p > 0 on T - U. Here is used to represent p_{(t}_i) in the paper. (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
Theorem | stoweidlem16 27134* | Lemma for stoweid 27181. The subset of functions in the algebra , with values in [ 0 , 1 ], is closed under multiplication. (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
Theorem | stoweidlem17 27135* | This lemma proves that the function (as defined in [BrosowskiDeutsh] p. 91, at the end of page 91) belongs to the subalgebra. (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
Theorem | stoweidlem18 27136* | This theorem proves Lemma 2 in [BrosowskiDeutsh] p. 92 when A is empty, the trivial case. Here D is used to denote the set A of Lemma 2, because the variable A is used for the subalgebra. (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
Theorem | stoweidlem19 27137* | If a set of real functions is closed under multiplication and it contains constants, then it is closed under finite exponentiation. (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
Theorem | stoweidlem20 27138* | If a set A of real functions from a common domain T is closed under the sum of two functions, then it is closed under the sum of a finite number of functions, indexed by G. (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
Theorem | stoweidlem21 27139* | Once the Stone Weierstrass theorem has been proven for approximating nonnegative functions, then this lemma is used to extend the result to functions with (possibly) negative values. (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
Theorem | stoweidlem22 27140* | If a set of real functions from a common domain is closed under addition, multiplication and it contains constants, then it is closed under subtraction. (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
Theorem | stoweidlem23 27141* | This lemma is used to prove the existence of a function p_{t} as in the beginning of Lemma 1 [BrosowskiDeutsh] p. 90: for all t in T - U, there exists a function p in the subalgebra, such that p_{t} ( t_{0} ) = 0 , p_{t} ( t ) > 0, and 0 <= p_{t} <= 1. (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
Theorem | stoweidlem24 27142* | This lemma proves that for sufficiently large, q_{n}( t ) > ( 1 - epsilon ), for all in : see Lemma 1 [BrosowskiDeutsh] p. 90, (at the bottom of page 90). is used to represent q_{n} in the paper, to represent in the paper, to represent , to represent δ, and to represent ε. (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
Theorem | stoweidlem25 27143* | This lemma proves that for n sufficiently large, q_{n}( t ) < ε, for all in : see Lemma 1 [BrosowskiDeutsh] p. 91 (at the top of page 91). is used to represent q_{n} in the paper, to represent n in the paper, to represent k, to represent δ, to represent p, and to represent ε. (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
Theorem | stoweidlem26 27144* | This lemma is used to prove that there is a function as in the proof of [BrosowskiDeutsh] p. 92: this lemma proves that g(t) > ( j - 4 / 3 ) * ε. Here is used to represnt j in the paper, is used to represent A in the paper, is used to represent t, and is used to represent ε. (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
Theorem | stoweidlem27 27145* | This lemma is used to prove the existence of a function p as in Lemma 1 [BrosowskiDeutsh] p. 90: p is in the subalgebra, such that 0 <= p <= 1, p_{(t}_0) = 0, and p > 0 on T - U. Here is used to represent p_{(t}_i) in the paper. (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
Theorem | stoweidlem28 27146* | There exists a δ as in Lemma 1 [BrosowskiDeutsh] p. 90: 0 < delta < 1 and p >= delta on . Here is used to represent δ in the paper. (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
Theorem | stoweidlem29 27147* | When the hypothesis for the extreme value theorem hold, then the inf of the range of the function belongs to the range, it is real and it a lower bound of the range. (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
Theorem | stoweidlem30 27148* | This lemma is used to prove the existence of a function p as in Lemma 1 [BrosowskiDeutsh] p. 90: p is in the subalgebra, such that 0 <= p <= 1, p_{(t}_0) = 0, and p > 0 on T - U. Z is used for t_{0}, P is used for p, is used for p_{(t}_i). (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
Theorem | stoweidlem31 27149* | This lemma is used to prove that there exists a function x as in the proof of Lemma 2 in [BrosowskiDeutsh] p. 91: assuming that is a finite subset of , indexes a finite set of functions in the subalgebra (of the Stone Weierstrass theorem), such that for all ranging in the finite indexing set, 0 ≤ x_{i} ≤ 1, x_{i} < ε / m on V(t_{i}), and x_{i} > 1 - ε / m on . Here M is used to represent m in the paper, is used to represent ε in the paper, v_{i} is used to represent V(t_{i}). (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
Theorem | stoweidlem32 27150* | If a set A of real functions from a common domain T is a subalgebra and it contains constants, then it is closed under the sum of a finite number of functions, indexed by G and finally scaled by a real Y. (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
Theorem | stoweidlem33 27151* | If a set of real functions from a common domain is closed under addition, multiplication and it contains constants, then it is closed under subtraction. (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
Theorem | stoweidlem34 27152* | This lemma proves that for all in there is a as in the proof of [BrosowskiDeutsh] p. 91 (at the bottom of page 91 and at the top of page 92): (j-4/3) * ε < f(t) <= (j-1/3) * ε , g(t) < (j+1/3) * ε, and g(t) > (j-4/3) * ε. Here is used to represent ε in the paper. (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
Theorem | stoweidlem35 27153* | This lemma is used to prove the existence of a function p as in Lemma 1 of [BrosowskiDeutsh] p. 90: p is in the subalgebra, such that 0 <= p <= 1, p_{(t}_0) = 0, and p > 0 on T - U. Here is used to represent p_{(t}_i) in the paper. (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
Theorem | stoweidlem36 27154* | This lemma is used to prove the existence of a function p_{t} as in Lemma 1 of [BrosowskiDeutsh] p. 90 (at the beginning of Lemma 1): for all t in T - U, there exists a function p in the subalgebra, such that p_{t} ( t_{0} ) = 0 , p_{t} ( t ) > 0, and 0 <= p_{t} <= 1. Z is used for t_{0} , S is used for t e. T - U , h is used for p_{t} . G is used for (h_{t})^2 and the final h is a normalized version of G ( divided by it's norm, see the variable N ). (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
Theorem | stoweidlem37 27155* | This lemma is used to prove the existence of a function p as in Lemma 1 of [BrosowskiDeutsh] p. 90: p is in the subalgebra, such that 0 <= p <= 1, p_{(t}_0) = 0, and p > 0 on T - U. Z is used for t_{0}, P is used for p, is used for p_{(t}_i). (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
Theorem | stoweidlem38 27156* | This lemma is used to prove the existence of a function p as in Lemma 1 of [BrosowskiDeutsh] p. 90: p is in the subalgebra, such that 0 <= p <= 1, p_{(t}_0) = 0, and p > 0 on T - U. Z is used for t_{0}, P is used for p, is used for p_{(t}_i). (Contributed by GlaucoSiliprandi, 20-Apr-2017.) |
Theorem | stoweidlem39 27157* | This lemma is used to prove that there exists a function x as in the proof of Lemma 2 in [BrosowskiDeutsh] p. 91: assuming that is a finite subset of , indexes a finite set of functions in the subalgebra (of the Stone Weierstrass theorem), such that for all i ranging in the finite indexing set, 0 ≤ x_{i} ≤ 1, x_{i} < ε / m on V(t_{i}), and x_{i} > 1 - ε / m on . Here is used to represent A in the paper's Lemma 2 (because is used for the subalgebra), is used to represent m in the paper, is used to represent ε, and v_{i} is used to represent V(t_{i}). is just a local definition, used to shorten statements. (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
Theorem | stoweidlem40 27158* | This lemma proves that q_{n} is in the subalgebra, as in the prove of Lemma 1 in [BrosowskiDeutsh] p. 90. Q is used to represent q_{n} in the paper, N is used to represent n in the paper, and M is used to represent k^n in the paper. (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
Theorem | stoweidlem41 27159* | This lemma is used to prove that there exists x as in Lemma 1 of [BrosowskiDeutsh] p. 90: 0 <= x(t) <= 1 for all t in T, x(t) < epsilon for all t in V, x(t) > 1 - epsilon for all t in T \ U. Here we prove the very last step of the proof of Lemma 1: "The result follows from taking x = 1 - q_{n"};. Here is used to represent ε in the paper, and to represent q_{n} in the paper. (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
Theorem | stoweidlem42 27160* | This lemma is used to prove that built as in Lemma 2 of [BrosowskiDeutsh] p. 91, is such that x > 1 - ε on B. Here is used to represent in the paper, and E is used to represent ε in the paper. (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
Theorem | stoweidlem43 27161* | This lemma is used to prove the existence of a function p_{t} as in Lemma 1 of [BrosowskiDeutsh] p. 90 (at the beginning of Lemma 1): for all t in T - U, there exists a function p_{t} in the subalgebra, such that p_{t}( t_{0} ) = 0 , p_{t} ( t ) > 0, and 0 <= p_{t} <= 1. Hera Z is used for t_{0} , S is used for t e. T - U , h is used for p_{t}. (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
Theorem | stoweidlem44 27162* | This lemma is used to prove the existence of a function p as in Lemma 1 of [BrosowskiDeutsh] p. 90: p is in the subalgebra, such that 0 <= p <= 1, p_{(t}_0) = 0, and p > 0 on T - U. Z is used to represent t_{0} in the paper. (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
Theorem | stoweidlem45 27163* | This lemma proves that, given an appropriate (in another theorem we prove such a exists), there exists a function q_{n} as in the proof of Lemma 1 in [BrosowskiDeutsh] p. 91 ( at the top of page 91): 0 <= q_{n} <= 1 , q_{n} < ε on T \ U, and q_{n} > 1 - ε on . We use y to represent the final q_{n} in the paper (the one with n large enough), to represent in the paper, to represent , to represent δ, to represent ε, and to represent . (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
Theorem | stoweidlem46 27164* | This lemma proves that sets U(t) as defined in Lemma 1 of [BrosowskiDeutsh] p. 90, are a cover of T \ U. Using this lemma, in a later theorem we will prove that a finite subcover exists. (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
Theorem | stoweidlem47 27165* | Subtracting a constant from a real continuous function gives another continuous function. (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
Theorem | stoweidlem48 27166* | This lemma is used to prove that built as in Lemma 2 of [BrosowskiDeutsh] p. 91, is such that x < ε on . Here is used to represent in the paper, is used to represent ε in the paper, and is used to represent in the paper (because is always used to represent the subalgebra). (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
Theorem | stoweidlem49 27167* | There exists a function q_{n} as in the proof of Lemma 1 in [BrosowskiDeutsh] p. 91 (at the top of page 91): 0 <= q_{n} <= 1 , q_{n} < ε on , and q_{n} > 1 - ε on . Here y is used to represent the final q_{n} in the paper (the one with n large enough), represents in the paper, represents , represents δ, represents ε, and represents . (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
Theorem | stoweidlem50 27168* | This lemma proves that sets U(t) as defined in Lemma 1 of [BrosowskiDeutsh] p. 90, contain a finite subcover of T \ U. (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
Theorem | stoweidlem51 27169* | There exists a function x as in the proof of Lemma 2 in [BrosowskiDeutsh] p. 91. Here is used to represent in the paper, because here is used for the subalgebra of functions. is used to represent ε in the paper. (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
Theorem | stoweidlem52 27170* | There exists a neighborood V as in Lemma 1 of [BrosowskiDeutsh] p. 90. Here Z is used to represent t_{0} in the paper, and v is used to represent V in the paper. (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
Theorem | stoweidlem53 27171* | This lemma is used to prove the existence of a function p as in Lemma 1 of [BrosowskiDeutsh] p. 90: p is in the subalgebra, such that 0 <= p <= 1, p_{(t}_0) = 0, and p > 0 on T - U. (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
Theorem | stoweidlem54 27172* | There exists a function as in the proof of Lemma 2 in [BrosowskiDeutsh] p. 91. Here is used to represent in the paper, because here is used for the subalgebra of functions. is used to represent ε in the paper. (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
Theorem | stoweidlem55 27173* | This lemma proves the existence of a function p as in the proof of Lemma 1 in [BrosowskiDeutsh] p. 90: p is in the subalgebra, such that 0 <= p <= 1, p_{(t}_0) = 0, and p > 0 on T - U. Here Z is used to represent t_{0} in the paper. (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
Theorem | stoweidlem56 27174* | This theorem proves Lemma 1 in [BrosowskiDeutsh] p. 90. Here is used to represent t_{0} in the paper, is used to represent in the paper, and is used to represent ε (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
Theorem | stoweidlem57 27175* | There exists a function x as in the proof of Lemma 2 in [BrosowskiDeutsh] p. 91. In this theorem, it is proven the non trivial case (the closed set D is nonempty). Here D is used to represent A in the paper, because the variable A is used for the subalgebra of functions. (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
Theorem | stoweidlem58 27176* | This theorem proves Lemma 2 in [BrosowskiDeutsh] p. 91. Here D is used to represent the set A of Lemma 2, because here the variable A is used for the subalgebra of functions. (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
Theorem | stoweidlem59 27177* | This lemma proves that there exists a function as in the proof in [BrosowskiDeutsh] p. 91, after Lemma 2: x_{j} is in the subalgebra, 0 <= x_{j} <= 1, x_{j} < ε / n on A_{j} (meaning A in the paper), x_{j} > 1 - \epslon / n on B_{j}. Here is used to represent A in the paper (because A is used for the subalgebra of functions), is used to represent ε. (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
Theorem | stoweidlem60 27178* | This lemma proves that there exists a function g as in the proof in [BrosowskiDeutsh] p. 91 (this parte of the proof actually spans through pages 91-92): g is in the subalgebra, and for all in , there is a such that (j-4/3)*ε < f(t) <= (j-1/3)*ε and (j-4/3)*ε < g(t) < (j+1/3)*ε. Here is used to represent f in the paper, and is used to represent ε. (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
Theorem | stoweidlem61 27179* | This lemma proves that there exists a function as in the proof in [BrosowskiDeutsh] p. 92: is in the subalgebra, and for all in , abs( f(t) - g(t) ) < 2*ε. Here is used to represent f in the paper, and is used to represent ε. For this lemma there's the further assumption that the function to be approximated is nonnegative (this assumption is removed in a later theorem). (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
Theorem | stoweidlem62 27180* | This theorem proves the Stone Weierstrass theorem for the non trivial case in which T is nonempty. The proof follows [BrosowskiDeutsh] p. 89 (through page 92). (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
Theorem | stoweid 27181* | This theorem proves the Stone-Weierstrass theorem for real valued functions: let be a compact topology on , and be the set of real continuous functions on . Assume that is a subalgebra of (closed under addition and multiplication of functions) containing constant functions and discriminating points (if and are distinct points in , then there exists a function in such that h(r) is distinct from h(t) ). Then, for any continuous function and for any positive real , there exists a function in the subalgebra , such that approximates up to ( represents the usual ε value). As a classical example, given any a,b reals, the closed interval could be taken, along with the subalgebra of real polynomials on , and then use this theorem to easily prove that real polynomials are dense in the standard metric space of continuous functions on . The proof and lemmas are written following [BrosowskiDeutsh] p. 89 (through page 92). Some effort is put in avoiding the use of the axiom of choice. (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
Theorem | stowei 27182* | This theorem proves the Stone-Weierstrass theorem for real valued functions: let be a compact topology on , and be the set of real continuous functions on . Assume that is a subalgebra of (closed under addition and multiplication of functions) containing constant functions and discriminating points (if and are distinct points in , then there exists a function in such that h(r) is distinct from h(t) ). Then, for any continuous function and for any positive real , there exists a function in the subalgebra , such that approximates up to ( represents the usual ε value). As a classical example, given any a,b reals, the closed interval could be taken, along with the subalgebra of real polynomials on , and then use this theorem to easily prove that real polynomials are dense in the standard metric space of continuous functions on . The proof and lemmas are written following [BrosowskiDeutsh] p. 89 (through page 92). Some effort is put in avoiding the use of the axiom of choice. The deduction version of this theorem is stoweid 27181: often times it will be better to use stoweid 27181 in other proofs (but this version is probably easier to be read and understood). (Contributed by Glauco Siliprandi, 20-Apr-2017.) |