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It is an old result of Klee saying that the infinite-dimensional Hilbert space is homeomorphic with both its unit sphere and its closed unit ball. See e.g. http://www.ams.org/journals/bull/1961-67-03/ …

Alternatively, you may use the Borsuk-Ulam antipodal theorem, in order to prove that such a map cannot be one-to-one.

Just another example. Consider the Banach space $\ell^{1}\left(\Gamma\right)$ , $\Gamma$ being an infinite set. Then the weak topology and the norm topology have the same convergent sequences (Schur' …

Is it true that, if S is a subring of a separable topological Noetherian ring R, then S is separable, too ?

Well, here is [most probably another] Wikipedia page: http://en.wikipedia.org/wiki/Cauchy-continuous_function. HTH. Alternatively, you may use the function $f:\mathbb{R\rightarrow\mathbb{R}}$ , expre …

First of all, the notion of $cone$ is a purely algebraic stuff, and not a metrical one. The $cone$ is naturally defined in the framework of $linear$ spaces, and not of Banach spaces. One can introduce …

Q2) has a negative answer. Namely, if, e.g., $g(x)=-x$ for all $x\in\mathbb{R}$, then there is no continuous $f:\mathbb{R\rightarrow\mathbb{R}}$ such that $f\circ f=g$. As to Q3, see, e.g., Theor …

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

Let m, n > 1. Is it true that C(S^m, S^m), and C(S^n, S^n) are homeomorphic ? [both endowed with the sup metric (or equivalently the compact-open topology)] Generally, C(S^n, S^n), with n >= 1, is a …