Two Remarks on Graph Norms

For a graph H, its homomorphism density in graphs naturally extends to the space of two-variable symmetric functions W in $$L^p$$ L p , $$p\ge e(H)$$ p ≥ e ( H ) , denoted by t(H, W). One may then define corresponding functionals $$\Vert W\Vert _{H}\,{:}{=}\,|t(H,W)|^{1/e(H)}$$ ‖ W ‖ H : = | t ( H , W ) | 1 / e ( H ) and $$\Vert W\Vert _{r(H)}\,{:}{=}\,t(H,|W|)^{1/e(H)}$$ ‖ W ‖ r ( H ) : = t ( H , | W | ) 1 / e ( H ) , and say that H is (semi-)norming if $$\Vert \,{\cdot }\,\Vert _{H}$$ ‖ · ‖ H is a (semi-)norm and that H is weakly norming if $$\Vert \,{\cdot }\,\Vert _{r(H)}$$ ‖ · ‖ r ( H ) is a norm. We obtain two results that contribute to the theory of (weakly) norming graphs. Firstly, answering a question of Hatami, who estimated the modulus of convexity and smoothness of $$\Vert \,{\cdot }\,\Vert _{H}$$ ‖ · ‖ H , we prove that $$\Vert \,{\cdot }\,\Vert _{r(H)}$$ ‖ · ‖ r ( H ) is neither uniformly convex nor uniformly smooth, provided that H is weakly norming. Secondly, we prove that every graph H without isolated vertices is (weakly) norming if and only if each component is an isomorphic copy of a (weakly) norming graph. This strong factorisation result allows us to assume connectivity of H when studying graph norms. In particular, we correct a negligence in the original statement of the aforementioned theorem by Hatami.

Strict and uniform convexity

Ways to classify and measure convexity of balls are described. Properties like strict convexity, uniform convexity, and squareness are discussed. The main tool, the modulus of convexity of a space, is studied. In the case of uniformly convex spaces, nearest point projections and asymptotic centres of sequences are presented.

On Generalized Moduli of Quasi-Banach Space

We shall discuss three generalized moduli such as generalized modulus of convexity, modulus of smoothness, and modulus of Zou-Cui of quasi-Banach spaces and give some important properties of these moduli. Furthermore, we establish relationships of these generalized moduli with each other.

Some Topological and Geometric Properties of Some New Spaces ofλ-Convergent and Bounded Series

The main purpose of this study is to introduce the spacescsλ,cs0λ, andbsλwhich areBK-spaces of nonabsolute type. We prove that these spaces are linearly isomorphic to the spacescs,cs0, andbs, respectively, and derive some inclusion relations. Additionally, Schauder bases of the spacescsλandcs0λhave been constructed and theα-,β-, andγ-duals of these spaces have been computed. Besides, we characterize some matrix classes from the spacescsλ,cs0λ, andbsλto the spaceslp,c, andc0, where1≤p≤∞. Finally, we examine some geometric properties of these spaces as Gurarǐ’s modulus of convexity, propertym∞, property(M), property WORTH, nonstrict Opial property, and weak fixed point property.

Banach-Saks Type and Gurariǐ Modulus of Convexity of Some Banach Sequence Spaces

Banach-Saks type is calculated for two types of Banach sequence spaces and Gurariǐ modulus of convexity is estimated from above for the spaces of one type among them.

Some New Generalized Difference Spaces of Nonabsolute Type Derived from the Spacesℓpandℓ∞

We introduce the sequence spaceℓpλ(B)of none absolute type which is ap-normed space andBKspace in the cases0<p<1and1⩽p⩽∞, respectively, and prove thatℓpλ(B)andℓpare linearly isomorphic for0<p⩽∞. Furthermore, we give some inclusion relations concerning the spaceℓpλ(B)and we construct the basis for the spaceℓpλ(B), where1⩽p<∞. Furthermore, we determine the alpha-, beta- and gamma-duals of the spaceℓpλ(B)for1⩽p⩽∞. Finally, we investigate some geometric properties concerning Banach-Saks typepand give Gurarii's modulus of convexity for the normed spaceℓpλ(B).