Riemann integration of continuous curves in a complete normed space, with the fundamental theorem of calculus and coefficient functionals

Since $[a,b]$ is compact and metric, hence a uniform space, [F3] provides a modulus: for every $\epsilon >0$ there is $\delta >0$ such that $\mathrm{\parallel }g(s)-g(s^{\mathrm{\prime }})\mathrm{\parallel }\le \epsilon$ whenever $\mathrm{\mid }s-s^{\mathrm{\prime }}\mathrm{\mid }\le \delta$. Define $\omega (\delta ):=\sup \{\mathrm{\parallel }g(s)-g(s^{\mathrm{\prime }})\mathrm{\parallel }:s,s^{\mathrm{\prime }}\in [a,b],\mathrm{\mid }s-s^{\mathrm{\prime }}\mathrm{\mid }\le \delta \}$; then $\omega (\delta )\to 0$ as $\delta \downarrow 0$ and $\omega$ is nondecreasing.

Let $P$ be a tagged partition $a=t_0<\cdots <t_n=b$ with tags $s_i\in [t_i,t_{i+1}]$ and Riemann sum $S_P:={\sum }_{i<n}(t_{i+1}-t_i)g(s_i)$, and let $R$ be a refinement of $P$ with any tags. Grouping the terms of $S_R$ by the interval of $P$ that contains them, $S_P-S_R={\sum }_{i<n}{\sum }_{R\ni [t^{\mathrm{\prime }},t^{\mathrm{\prime }\mathrm{\prime }}]\subseteq [t_i,t_{i+1}]}(t^{\mathrm{\prime }\mathrm{\prime }}-t^{\mathrm{\prime }})(g(s_i)-g(s^{\mathrm{\prime }}))$, so $\mathrm{\parallel }{S}_P-S_R\mathrm{\parallel }\le {\sum }_{i<n}(t_{i+1}-t_i)\omega (t_{i+1}-t_i)\le (b-a)\omega (\mathrm{m}\mathrm{e}\mathrm{s}\mathrm{h}(P))$ by [F1].

Let $(P^{(j)})$ and $(Q^{(j)})$ be two sequences of tagged partitions with meshes tending to $0$. For each $j$ let $R^{(j)}$ be a common refinement of $P^{(j)}$ and $Q^{(j)}$, so $\mathrm{m}\mathrm{e}\mathrm{s}\mathrm{h}(R^{(j)})\le \min (\mathrm{m}\mathrm{e}\mathrm{s}\mathrm{h}(P^{(j)}),\mathrm{m}\mathrm{e}\mathrm{s}\mathrm{h}(Q^{(j)}))$. By step 2.1 applied twice, $\mathrm{\parallel }{S}_{P^{(j)}}-S_{Q^{(j)}}\mathrm{\parallel }\le \mathrm{\parallel }{S}_{P^{(j)}}-S_{R^{(j)}}\mathrm{\parallel }+\mathrm{\parallel }{S}_{R^{(j)}}-S_{Q^{(j)}}\mathrm{\parallel }\le 2(b-a)\omega (\max (\mathrm{m}\mathrm{e}\mathrm{s}\mathrm{h}(P^{(j)}),\mathrm{m}\mathrm{e}\mathrm{s}\mathrm{h}(Q^{(j)})))\to 0$. For two indices $j,k$ the same common-refinement estimate gives $\mathrm{\parallel }{S}_{P^{(j)}}-S_{P^{(k)}}\mathrm{\parallel }\le 2(b-a)\omega (\max (\mathrm{m}\mathrm{e}\mathrm{s}\mathrm{h}(P^{(j)}),\mathrm{m}\mathrm{e}\mathrm{s}\mathrm{h}(P^{(k)})))\to 0$ as both indices tend to infinity, so $(S_{P^{(j)}})$ is Cauchy, and taking arbitrary sequences shows that any two such limits coincide; by completeness [F2] the limit ${\int }_a^{b}g$ exists and is independent of the sequence. Moreover $\mathrm{\parallel }{S}_P\mathrm{\parallel }\le (b-a){\sup }_{[a,b]}\mathrm{\parallel }g\mathrm{\parallel }$ for every $P$ by the triangle inequality, and the norm is continuous [F1], so the same bound holds for the limit. This is claim 1.

For $a\le c\le b$ the integral over $[a,b]$ equals the sum of those over $[a,c]$ and $[c,b]$: if $c=a$ or $c=b$ this is the zero-integral convention, and otherwise partitions of $[a,b]$ may be refined to contain $c$, and for such a partition the Riemann sum splits into the two sums; both sides converge and limits are additive because addition in $X$ is continuous [F1]. Defining $G(t):={\int }_a^{t}g$, this additivity gives $G(t)-G(s)={\int }_s^{t}g$ for $a\le s\le t\le b$, and claim 1 applied on $[s,t]$ gives $\mathrm{\parallel }G(t)-G(s)\mathrm{\parallel }\le (t-s){\sup }_{[s,t]}\mathrm{\parallel }g\mathrm{\parallel }$.

Linearity: for fixed $g,h,\alpha ,\beta$ and any tagged partition $P$, the Riemann sums satisfy $S_P(\alpha g+\beta h)=\alpha S_P(g)+\beta S_P(h)$ by finite distributivity in the vector space $X$ [F1]; letting the mesh tend to $0$ and using continuity of addition and scalar multiplication [F1] gives $\int (\alpha g+\beta h)=\alpha \int g+\beta \int h$. This completes claim 3.

Differentiability: for $a\le t<b$ and $0<h\le b-t$, step 4.1 gives $G(t+h)-G(t)={\int }_t^{t+h}g$, and claim 1 gives $\parallel \frac{1}{h}(G(t+h)-G(t))-g(t)\parallel =\parallel \frac{1}{h}{\int }_t^{t+h}(g-g(t))\parallel \le {\sup }_{u\in [t,t+h]}\mathrm{\parallel }g(u)-g(t)\parallel \to 0$ as $h\downarrow 0$ by continuity [F4]; for $h<0$ with $a\le t+h<t$, oriented additivity gives the same quotient estimate with supremum over $[t+h,t]$, so $G^{\mathrm{\prime }}=g$ with the usual one-sided interpretation at $a$ and $b$. Together with step 4.1 this is claim 2.

Bounded linear images: $T\circ g$ is continuous since $g$ is and $T$ is Lipschitz [F1]; for each tagged partition $T(S_P(g))=S_P(T\circ g)$ by linearity, and $T$ is continuous, so passing to the limit in step 3.1 gives $T({\int }_a^{b}g)={\int }_a^{b}T\circ g$. Applying this to a bounded linear functional $\mathrm{\Lambda }:X\to \mathbb{R}$ (with $\mathbb{R}$ in the role of $Y$) gives the last clause of claim 4.