fa.functional analysis - Is this a $C^{infty}$ function ?

Here's an attempt at the second part. I'm much more familiar with the convenient calculus of Frölicher, Kriegl, and Michor than with Fréchet derivatives, but for Hilbert spaces, there shouldn't be too much in it.

Let $rho colon mathbb{R} to mathbb{R}$ be a smooth bump function at $0$ with support in $[-1/2,1/2]$. Since $|e_n - e_m| = sqrt{2}$ (for $n ne m$), if we define $f_n colon H to mathbb{R}$ by $f_n(x) = rho(|e_n - x|^2)$ then the $f_n$s have disjoint support. Then $f := sum n f_n colon H to mathbb{R}$ is locally smooth and hence smooth. In addition, $f(e_n) = n$ so $f$ maps the unit ball onto an unbounded set.

(How does that look? I'm well aware that I may have overlooked something really obvious!)

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Edit 2012-11-12: I did overlook something that I shouldn't have done. The argument above is not quite correct: that the $f_n$s have disjoint support is not enough to know that the sum $sum n f_n$ is smooth. For that, I need to know that the $f_n$s have locally finite support. Fortunately, this is true. If $f_n(x) ne 0$ then $|e_n - x| < 1/4$. Thus for $y in H$ consider the ball of radius $1/4$ about $y$. If we have $x_1$ and $x_2$ in this ball and $n,m$ such that $f_n(x_1) ne 0$ and $f_m(x_2) ne 0$ then $|e_n - e_m| le |e_n - x_1| + |x_1 - y| + |y - x_2| + |x_2 - e_m| < 1$ whence $n = m$. Hence at most one $f_n$ has support intersecting this ball about $y$ and so the supports of the $f_n$ are locally finite.

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