Strict U-Ideals and U-Summands in Banach Spaces

For a strict u-ideal $X$ in a Banach space $Y$ we show that the set of points in the dual unit ball $B_{X^{\ast}}$, strongly exposed by points in the range $\it TY$ of the unconditional extension operator $T$ from $Y$ into the bidual $X^{\ast\ast}$ of $X$, is contained in the weak$^{\ast}$ denting points in $B_{X^{\ast}}$. We also prove that a u-embedded space is a u-summand if and only if it contains no copy of $c_0$ if and only if it is weakly sequentially complete.

There are no denting points in the unit ball of 𝒫(2H)

For any infinite dimensional real Hilbert space H we show that the unit ball of the space of continuous 2-homogeneous polynomials on H, 𝒫(2H), has no denting points. Thus the unit ball of 𝒫(2H) has no strongly exposed points.