Strictly convex space

Banach Spaces

Published in Hugo D. Junghenn, Principles of Analysis, 2018

A Euclidean space is an example of a strictly convex space. By contrast, Rd $ \mathbb R ^{d} $ with the norm ‖x‖1=∑j=1d|xj| $ \Vert {\boldsymbol{x}}\Vert _1 = \sum _{j = 1}^d |x_j| $ or ‖x‖∞=max1≤j≤d|xj| $ \Vert {\boldsymbol{x}}\Vert _\infty = \max _{1 \le j \le d} |x_j| $ is not strictly convex. A more interesting example is Lp(X,F,μ) $ L^p(X,\mathcal{F},\mu ) $ . If 1<p<∞ $ 1 < p < \infty $ , then Lp $ L^p $ is strictly convex by the second part of 4.1.3. On the other hand, if p=1 $ p = 1 $ or ∞ $ \infty $ , then Lp $ L^p $ is strictly convex only in trivial circumstances (Ex. 8.23).

Bishop–Phelps cones given by an equation in Banach spaces

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Journal Information

Published in Optimization, 2023

Truong Xuan Duc Ha, Johannes Jahn

All Hilbert spaces, the Banach spaces and () are strictly convex; but the Banach spaces and (p = 1 or ), are not strictly convex. Klee (compare [21]) made the conjecture that every reflexive Banach space is isomorphic to a strictly convex space. Clarkson [20, Theorem 9] showed that in every separable Banach space a new norm being equivalent to the original norm may be given, with respect to which the space is strictly convex. There are many papers investigating and characterizing strictly convex normed spaces (e.g. see [22–27]).

Strictly convex space

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Banach Spaces

Bishop–Phelps cones given by an equation in Banach spaces