I'm currently working on fixed point theorems on uniformly convex spaces and I will love if anyone can point my attention to spaces that are uniformly convex apart from the ones I have listed above.
There are many others, because several standard operations with Banach spaces (like passing to subspaces, quotient spaces, direct sums with "good" norms) preserve this property. So, you can construct many new uniformly convex spaces starting from the "elementary" ones. For example, $L_3[0, 1] \oplus_2 \ell_4$ is uniformly convex. Also, many spaces of vector-valued functions enjoy this property. The most important result in the theory: a Banach space can be equivalently renormed to be a uniformly convex space if and only if it is superreflexive. "Natural" examples of uniformly convex spaces you will find in the theory of Orlicz spaces.
There are many others, because several standard operations with Banach spaces (like passing to subspaces, quotient spaces, direct sums with "good" norms) preserve this property. So, you can construct many new uniformly convex spaces starting from the "elementary" ones. For example, $L_3[0, 1] \oplus_2 \ell_4$ is uniformly convex. Also, many spaces of vector-valued functions enjoy this property. The most important result in the theory: a Banach space can be equivalently renormed to be a uniformly convex space if and only if it is superreflexive. "Natural" examples of uniformly convex spaces you will find in the theory of Orlicz spaces.
@Vladimir Kadets
Thank you so much for your response.
I was looking for a more "general (examples) name" of uniformly convex spaces.
A pre-Hilbert space (an incomplete space with inner product ) is an uniformly convex space. See:
K. Yosida, Functional Analysis, 1965.
I may direct you to a lovely (almost) classical book by Joseph Diestel, Sequences and Series in Banach spaces, Springer Verlag, 1984, ISBN 0-387-90859-5. Look at the material and the author's comments therein. The bibliography will be of a great help, as well.
Another, more recent item is Topics in Banach Space Theory by Fernando Albiac and Nigel J. Kalton, Springer-Verlag, ISBN 978-7-5100-4804-3. Here, the treatment of u.c. Banach spaces is more succint.
Good luck in studying this beautiful subject!
Roman Sznajder , thank you very much sir. You have all been helpful, thank you all.
Thank you all for your contribution (my undergraduate project work). I felt fulfilled and happy going through my work, it is one of the best at the department.
@Roman Sznajder @N. M. Ratiner @Vladimir Kadets
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