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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 …

[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 an infinite dimensional Banach space, let $E^{\ast}$ denote its continuous (i.e., Banach space) dual, and let $E'$ be its algebraic dual. Clearly, $E^{\ast}$ is a proper vector subspace of …

Let $E$ be an arbitrary Banach space and let $T:E^{*}\rightarrow\ell^{2}$ be a linear continuous operator. Is it true that $T$ must be the $so$-limit (i.e., limit w.r.t. the strong operator topology) …

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 ${\mathcal{F}}$ : …

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 ? …

To 1) and 2). Let $X^{'}=Y=c_{0}$ , and let $X=c_{00}$. Take some $p\in X^{'}\smallsetminus X$, and define $T:X\rightarrow Y$ by $Tx:=\left\Vert x\right\Vert \cdot\left(\sin\left(\left\Vert x-p\right\ …

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 …

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

Let H be an infinite dimensional real Hilbert space. A [not necessarily linear] mapping of H into itself is said to be hemicontinuous if it is continuous from each line segment of H to the weak topol …

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 …

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

It seems that simply one can't measure it. Here below is briefly described an example of a nondegenerate indefinite inner product space having no cardinal-valued Morse index. Consider $(\ell^{2},<.,. …

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 …