# Function space between uniform continuity and Hölder continuity

Can you give an example of a complete metric vector space of uniformly continuous functions that is strictly contained between the set of uniformly continuous functions on $\mathbb R^d$ and the Hölder spaces on $\mathbb R^d$?

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Can you replace the function $|\cdot|^\alpha$ in the definition of Holder spaces by another function? –  timur Sep 11 '11 at 22:46
The phrasing of the question makes it sound unfortunately like something from an assignment. You might get more detailed responses if you gave more information about how you ran into this question, and gave some motivation (but also see the FAQ concerning what questions should be asked here and how best to formulate them) –  Yemon Choi Sep 12 '11 at 1:19
What do you mean by containment in this case? Contained as sets of functions? Must there be some kind of isometric vector space embedding-type relationship? If so, what are the topologies on each of the spaces? –  Peter Luthy Sep 12 '11 at 7:38
Thank you for the remarks. The question deliberately open, and solely motivated by curiousity, therefore i did not specify the nature of embedding. But thanks for the comment on the Hölder condition. FYI, Sobolev spaces with position-dependent exponents and integrability are considered in some parts of numerical analysis. –  shuhalo Sep 15 '11 at 4:01