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This is Problem 54 in Halmos' "A Hilbert Space Problem Book". However, I think this is a concrete counterexample. [Please let me know if not viewable.]

The answer is negative. For, pick some non-zero $e$ in $E$, and choose a surjection $\rho\in C\left(O,\mathbb{R}\right)$ (there exists !). Next, consider the (uncountable, uniformly discrete) family …

In 1974, Aharoni proved that every separable metric space (X, d) is Lipschitz isomorphic to a subset of the Banach space c_0. Thus, for some constant L, there is a map K: X --> c_0 that satisfies the …

This can be considered as a relative of Splitting a space into positive and negative parts. Is there a real (non-trivial) vector space $V$, endowed with a nondegenerate symmetric bilinear pairing $\l …

[In what follows $0^{0}$= 1 by convention.] Is there some closed infinite dimensional linear subspace $F$ of $L^{2}(0,1)$ such that $\left\lvert f\right\rvert^{\left\lvert f\right\rvert}$ belongs to $ …

Let $K$ be the closed unit ball of some infinite dimensional Banach space, and let $H$ be an autohomeomorphism of $K$, having fixed points. Can $H/2$ be fixed point free ? Also, let ${\mathcal{F}}$ : …

[Just a historical remark.] AFAIK, the fact that the set of all extreme points of a compact convex subset of $\mathbb{R}^{2}$ must be closed is due to the legendary American mathematician G. Baley Pri …

Is there a "right" definition of the nuclear operator in the nonlinear framework ? Of course, such an operator must be compact, while a linear operator should be "nonlinearly" nuclear iff it is nuclea …

Let { $ L_{n} $ } be the sequence of Laguerre polynomials, and let us define $e_{n}(t)=\dfrac{L_{n}(\ln t)}{t}$ $\cdot\chi_{\left(1,\infty\right)}\left(t\right)$ $(n\in\mathbb{N\textrm{, t > 0}})$ …

Is this little toy known ? Let $E$ be some Banach space, and let $K$ be the closed unit ball of its dual, endowed with the weak-star topology. Also, let $j:E$ $\rightarrow$ $C(K)$ be the natural embe …

The answer is negative. Since the linear span of the Dirac masses is not a dense subspace of the dual of $C[0,1]$.

This is a negative result, heavily relying on P. Dodos' answer to Boundedness of nonlinear continuous functionals. Let $\kappa$ be a measurable cardinal, let $K$ be the closed unit ball of $\ell_{\i …

I think that http://www.ams.org/journals/tran/1982-271-02/S0002-9947-1982-0654848-2/S0002-9947-1982-0654848-2.pdf [together with its references] provides us with several counterexamples [as well as wi …

Just two isometric/isomorphic characterizations: A Banach space $X$ is [isometric to] a Hilbert space if and only if there exists a Banach space $Y$ and a symmetric bilinear mapping $f:X\times X\righ …

Given $X$, $Y$ two real Banach spaces, let's say that $(X,\ Y)$ is a Borsuk pair if for any continuous mapping $T$ : {$x$ $\in$ $X$ ; $||x||\leq1$} $\rightarrow$ $Y$ s.t. $T$ is odd on {$x$ $\in$ $X$ …