This gives me a very nice excuse to talk about two of my favourite German math books of all time:
For one there is the three-volume book set Analysis (I-III) by Herbert Amann and Joachim Escher. These books were originally written in German, but were translated into English later on. These books are definitely one of the "definitive" undergrad analysis books in Germany. They encompass nearly everything that is taught in undergrad analysis classes, except functional analysis.
What makes them so great is their strict almost Bourbaki-style build-up of analysis: starting from basic logic predicates they establish everything one needs to know to get to topics like Stokes theorem or the Lebesgue integral. They cover:
Analysis I: This book starts from basic logic and "naivé" set theory and establishes the basic notions of rings, fields and vector spaces. Then the natural numbers and the real numbers are constructed which is followed by everything you'd espect in a first-semester analysis course: sequences, series, continuous functions on metric spaces and derivatives on the reals. The book concludes with sequences of functions.
Analysis II: The results about sequences of functions are swiftly put to use to construct the Riemann-Cauchy integral on Banach spaces. This is followed by an expansive treatise of differential calculus on Banach spaces including the implicit function theorem and the inverse function theorem. This theory is then applied to ordinary differential equations and variational calculus proving the Picard-Lindelöf-theorem and the Euler-Lagrange equations. The second part of the book is devoted to submanifolds of $\mathbb{R^d}$ and curve-integrals. This is applied to establish basic results in complex analysis.
Analysis III: Basic measure theory is introduced culminating in the Carathéodory extension theorem and the construction of the Lebesgue- and Hausdorff-measures. Then the (nunmerical) Lebesgue integral and the Bochner integral on Banach spaces is constructed with their standard convergence theorems and the transformation theorem. This is applied to $L^p$-spaces and the Fourier transformation. The second part of the book treats general submanifolds and the theory of differential forms on them. Integration of differential on submanifolds of $\mathbb{R^d}$ is defined using the Lebesgue integral. Then as the finalé of the series Stokes theorem is proved and the theory is applied to vector analysis.
Then there is the book Maß- und Integrationstheorie ("Measure- and Integration theory") by Jürgen Elstrodt. It seems that there is no english translation. What is very special about this book that it has many historical remarks and biographies of the mathematicians which contributed to the field.
The name of the book suggest what the book is about: It very thoroughly covers the theory of rings, pre-measures, measures, $\sigma$-algebras and all that. Various extension and uniqueness theorems are proved. This all is used to construct the abstract Lebesgue integral. Then the various notions of convergence one has in measure theory are introduced and at the end the theory of signed measures and measures on topological spaces including the Haar measure and the Riesz representation theorem is covered.
A honourable mention is Grundkurs Funktionalanalysis ("Basics of Functional Analysis") by Winfried Kaballo, again not translated into english. It is a very structured textooks with great explanations. It contains everything an undergrad course in functional analysis would be about: Banach- and Hilbert space theory, the "pillar"-theoreoms of functional analysis like Hahn-Banach and concludes with the spectal theory of self-adjpoint operators. The spectral theorem is proved for compact operators and a first glimpse on unbounded operators is given. It also has a continuation which is called Aufbaukurs Funktionalanalysis ("Advanced Functional Analysis"), but I've never read that.
