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- https://math.libretexts.org/Bookshelves/Analysis/Mathematical_Analysis_(Zakon)/02%3A_Real_Numbers_and_Fields/2.05%3A_Some_Consequences_of_the_Completeness_AxiomThis proves our last assertion and shows that noy∈F can be a right bound of N( for y<n∈N), or a left bound of J( for y>−m∈J).◻ Now, by Theorem 2 of \(§§5-6, A+...This proves our last assertion and shows that noy∈F can be a right bound of N( for y<n∈N), or a left bound of J( for y>−m∈J).◻ Now, by Theorem 2 of §§5-6, A+m has a minimum; call it p . As p is the least of all sums x+m, p-m is the least of all x \in A ; so p-m=\min A exists, as claimed.
- https://math.libretexts.org/Bookshelves/Analysis/Mathematical_Analysis_(Zakon)/05%3A_Differentiation_and_Antidifferentiation/5.04%3A_Complex_and_Vector-Valued_Functions_on_(E1)Let f^{\prime} \geq 0 on I-Q. Fix any x, y \in I(x<y) and define g(t)=0 on E^{1}. Then \left|g^{\prime}\right|=0 \leq f^{\prime} on I-Q . Thus g and f satisfy Theor...Let f^{\prime} \geq 0 on I-Q. Fix any x, y \in I(x<y) and define g(t)=0 on E^{1}. Then \left|g^{\prime}\right|=0 \leq f^{\prime} on I-Q . Thus g and f satisfy Theorem 1 (with their roles reversed on I, and certainly on the subinterval [x, y]. Thus we have
- https://math.libretexts.org/Bookshelves/Analysis/Mathematical_Analysis_(Zakon)/05%3A_Differentiation_and_Antidifferentiation/5.05%3A_Antiderivatives_(Primitives_Integrals)If F^{\prime}=f on a set B \subseteq I, we say that \int f is exact on B and call F an exact primitive on B. Thus if Q=\emptyset, \int f is exact on all of I. Thus, set...If F^{\prime}=f on a set B \subseteq I, we say that \int f is exact on B and call F an exact primitive on B. Thus if Q=\emptyset, \int f is exact on all of I. Thus, setting H=f g, we have H=\int\left(f g^{\prime}+f^{\prime} g\right) on I. Hence by Corollary 1 if \int f^{\prime} g exists on I, so does \int\left(\left(f g^{\prime}+f^{\prime} g\right)-f^{\prime} g\right)=\int f g^{\prime}, and
- https://math.libretexts.org/Bookshelves/Analysis/Mathematical_Analysis_(Zakon)/05%3A_Differentiation_and_Antidifferentiation/5.11%3A_Integral_Definitions_of_Some_Functions\[\begin{aligned} \log x y &=\int_{1}^{x y} \frac{1}{t} d t=\int_{1 / x}^{y} \frac{1}{s} d s \\ &=\int_{1 / x}^{1} \frac{1}{s} d s+\int_{1}^{y} \frac{1}{s} d s \\ &=-\log \frac{1}{x}+\log y \\ &=\log ...\begin{aligned} \log x y &=\int_{1}^{x y} \frac{1}{t} d t=\int_{1 / x}^{y} \frac{1}{s} d s \\ &=\int_{1 / x}^{1} \frac{1}{s} d s+\int_{1}^{y} \frac{1}{s} d s \\ &=-\log \frac{1}{x}+\log y \\ &=\log x+\log y. \end{aligned} The function F as redefined in Theorem 2 will be denoted by F_{0}. It is a primitive of f on the closed interval \overline{I} (exact on I). Thus F_{0}(x)=\int_{0}^{x} f, -1 \leq x \leq 1, and we may now write
- https://math.libretexts.org/Bookshelves/Analysis/Mathematical_Analysis_(Zakon)/07%3A_Volume_and_Measure/7.08%3A_Lebesgue_MeasureAs we saw in §§5 and 6, this premeasure induces an outer measure m^{*} on all subsets of E^{n}; and m^{*}, in turn, induces a measure m on the \sigma-field \mathcal{M}^{*} of \...As we saw in §§5 and 6, this premeasure induces an outer measure m^{*} on all subsets of E^{n}; and m^{*}, in turn, induces a measure m on the \sigma-field \mathcal{M}^{*} of m^{*}-measurable sets. More generally, a \sigma-finite set A \in \mathcal{M} in a measure space (S, \mathcal{M}, \mu) is a countable union of disjoint sets of finite measure (Corollary 1 of §1).
- https://math.libretexts.org/Bookshelves/Analysis/Mathematical_Analysis_(Zakon)/07%3A_Volume_and_Measure/7.02%3A_(mathcalC_sigma)-Sets._Countable_Additivity._Permutable_Series\[\begin{aligned} 2 \varepsilon=\sum_{i=1}^{m} v A_{i}-\sum_{k=1}^{\infty} v B_{k} &<\sum_{i=1}^{m}\left(v X_{i}+\frac{\varepsilon}{m}\right)-\sum_{k=1}^{\infty}\left(v Y_{k}-\frac{\varepsilon}{2^{k}}...\begin{aligned} 2 \varepsilon=\sum_{i=1}^{m} v A_{i}-\sum_{k=1}^{\infty} v B_{k} &<\sum_{i=1}^{m}\left(v X_{i}+\frac{\varepsilon}{m}\right)-\sum_{k=1}^{\infty}\left(v Y_{k}-\frac{\varepsilon}{2^{k}}\right) \\ &=\sum_{i=1}^{m} v X_{i}-\sum_{k=1}^{\infty} v Y_{k}+2 \varepsilon. \end{aligned} \sum_{n, k=1}^{\infty} a_{n k}=\sum_{n=1}^{\infty}\left(\sum_{k=1}^{\infty} a_{n k}\right)=\sum_{k=1}^{\infty}\left(\sum_{n=1}^{\infty} a_{n k}\right).
- https://math.libretexts.org/Bookshelves/Analysis/Mathematical_Analysis_(Zakon)/05%3A_Differentiation_and_Antidifferentiation/5.07%3A_The_Total_Variation_(Length)_of_a_Function_f_-_E1__E\[\begin{aligned}\left|\Delta_{i} h f\right| &=\left|h\left(t_{i}\right) f\left(t_{i}\right)-h\left(t_{i-1}\right) f\left(t_{i-1}\right)\right| \\ & \leq\left|h\left(t_{i}\right) f\left(t_{i}\right)-h...\[\begin{aligned}\left|\Delta_{i} h f\right| &=\left|h\left(t_{i}\right) f\left(t_{i}\right)-h\left(t_{i-1}\right) f\left(t_{i-1}\right)\right| \\ & \leq\left|h\left(t_{i}\right) f\left(t_{i}\right)-h\left(t_{i-1}\right) f\left(t_{i}\right)\right|+\left|h\left(t_{i-1}\right) f\left(t_{i}\right)-h\left(t_{i-1}\right) f\left(t_{i-1}\right)\right| \\ &
- https://math.libretexts.org/Bookshelves/Analysis/Mathematical_Analysis_(Zakon)/03%3A_Vector_Spaces_and_Metric_Spaces/3.03%3A_Intervals_in_E\[\begin{aligned}(\overline{a}, \overline{b}) &=\left\{\overline{x} | a_{k}<x_{k}<b_{k}, k=1,2, \ldots, n\right\} \\ &=\left(a_{1}, b_{1}\right) \times\left(a_{2}, b_{2}\right) \times \cdots \times\le...\[\begin{aligned}(\overline{a}, \overline{b}) &=\left\{\overline{x} | a_{k}<x_{k}<b_{k}, k=1,2, \ldots, n\right\} \\ &=\left(a_{1}, b_{1}\right) \times\left(a_{2}, b_{2}\right) \times \cdots \times\left(a_{n}, b_{n}\right) \\ [\overline{a}, \overline{b}] &=\left\{\overline{x} | a_{k} \leq x_{k} \leq b_{k}, k=1,2, \ldots, n\right\} \\ &=\left[a_{1}, b_{1}\right] \times\left[a_{2}, b_{2}\right] \times \cdots \times\left[a_{n}, b_{n}\right] \\ (\overline{a}, \overline{b}] &=\left\{\overline{x} | a…
- https://math.libretexts.org/Bookshelves/Analysis/Mathematical_Analysis_(Zakon)/03%3A_Vector_Spaces_and_Metric_Spaces/3.05%3A_Vector_Spaces._The_Space_C._Euclidean_Spaces\[\begin{aligned}\left|t x+y^{\prime}\right|^{2} &=\left(t x+y^{\prime}\right) \cdot\left(t x+y^{\prime}\right) \\ &=t x \cdot t x+y^{\prime} \cdot t x+t x \cdot y^{\prime}+y^{\prime} \cdot y^{\prime}...\begin{aligned}\left|t x+y^{\prime}\right|^{2} &=\left(t x+y^{\prime}\right) \cdot\left(t x+y^{\prime}\right) \\ &=t x \cdot t x+y^{\prime} \cdot t x+t x \cdot y^{\prime}+y^{\prime} \cdot y^{\prime} \\ &=t^{2}(x \cdot x)+t\left(y^{\prime} \cdot x\right)+t\left(x \cdot y^{\prime}\right)+\left(y^{\prime} \cdot y^{\prime}\right) \end{aligned}
- https://math.libretexts.org/Bookshelves/Analysis/Mathematical_Analysis_(Zakon)/03%3A_Vector_Spaces_and_Metric_Spaces/3.12%3A_More_on_Cluster_Points_and_Closed_Sets._DensityIf p \notin A, then p \in-A; therefore, by Definition 3 in §12, some G_{p} fails to meet A\left(G_{p} \cap A=\emptyset\right). Hence no p \in-A is a cluster point, or the limit of ...If p \notin A, then p \in-A; therefore, by Definition 3 in §12, some G_{p} fails to meet A\left(G_{p} \cap A=\emptyset\right). Hence no p \in-A is a cluster point, or the limit of a sequence \left\{x_{n}\right\} \subseteq A. (This would contradict Definitions 1 and 2 of §14.) Consequently, all such cluster points and limits must be in A, as claimed.
- https://math.libretexts.org/Bookshelves/Analysis/Mathematical_Analysis_(Zakon)/03%3A_Vector_Spaces_and_Metric_Spaces/3.06%3A_Normed_Linear_SpacesBy a normed linear space (briefly normed space) is meant a real or complex vector space E in which every vector x is associated with a real number |x|, called its absolute value or norm, i...By a normed linear space (briefly normed space) is meant a real or complex vector space E in which every vector x is associated with a real number |x|, called its absolute value or norm, in such a manner that the properties \left(\mathrm{a}^{\prime}\right)-\left(\mathrm{c}^{\prime}\right) of §9 hold.
