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The question below is related to the classical Browder-Goehde-Kirk fixed point theorem. Let $K$ be the closed unit ball of $\ell^{2}$, and let $T:K\rightarrow K$ be a mapping such that $$\Vert Tx-Ty\V …

The space of tempered distributions is sequential (for its usual strong topology). See, e.g., Dudley, and the references therein.

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

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 …

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 …

Is the following result true: the Hilbert space $\ell^{2}\left(2^{\Gamma}\right)$ is a quotient of $\ell^{\infty}\left(\Gamma\right)$ for any uncountable $\Gamma$ ? [I think it is, but cannot remember …

Let $K$ be the closed unit ball of $C[0,1]$, and let $f$ in $C(K,\mathbb{\, R})$. Is it true that there exists an infinite dimensional reflexive subspace $E$ of $C[0,1]$ s.t. $f(K\cap E)$ is bounded ? …

Let E be a normed space, and let $T$:E * $\rightarrow$ E * be a nonlinear operator. Suppose that : 1) $T$ is continuous from (E *, ||.||) to itself (i.e., it is norm-continuous). and 2) $T$ is co …

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 {$x$ $\in$ $X$ …

One may also remark that $c_{0}$ $\times$ (ℓ∞$/c_{0}$) is a predual of (ℓ∞)$^*$. However, $c_{0}$ $\times$ (ℓ∞$/c_{0}$) is not isomorphic to ℓ∞.

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 $E$ be a real linear space, endowed with a non-degenerate symmetric bilinear form $(.,.)$. Suppose that the [indefinite] inner product space $(E,(.,.))$ satisfies the following [sequential] prope …

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 …

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.]