  Functional data analysis is typically conducted within the $L^2$-Hilbert
space framework. There is by now a fully developed statistical toolbox allowing
for the principled application of the functional data machinery to real-world
problems, often based on dimension reduction techniques such as functional
principal component analysis. At the same time, there have recently been a
number of publications that sidestep dimension reduction steps and focus on a
fully functional $L^2$-methodology. This paper goes one step further and
develops data analysis methodology for functional time series in the space of
all continuous functions. The work is motivated by the fact that objects with
rather different shapes may still have a small $L^2$-distance and are therefore
identified as similar when using an $L^2$-metric. However, in applications it
is often desirable to use metrics reflecting the visualization of the curves in
the statistical analysis. The methodological contributions are focused on
developing two-sample and change-point tests as well as confidence bands, as
these procedures appear do be conducive to the proposed setting. Particular
interest is put on relevant differences; that is, on not trying to test for
exact equality, but rather for pre-specified deviations under the null
hypothesis.
  The procedures are justified through large-sample theory. To ensure
practicability, non-standard bootstrap procedures are developed and
investigated addressing particular features that arise in the problem of
testing relevant hypotheses. The finite sample properties are explored through
a simulation study and an application to annual temperature profiles.
