Doubt in definition of symmetric continuous function and norm in Kupka's paper

Question

In this article "Counterexample to the Morse-Sard theorem in the case of infinite-dimensional manifolds" of I. Kupka has the following passage:

For $H=l^{2}$

"Let $H^{*}$ be the dual of $H$. A base of $H^{*}$ is formed by the linear functions $e_{1},e_{2},...,e_{n},...$ where

$e_{n}(x)=x_{n}$ where $x= (x_{1},x_{2},...,x_{k},...)$

For any $k$, let $\sum_{k}(H^{*})$ be the space of all $k$-linear symmetric continuous functions on $H$. $\sum_{k}(H^{*})$ is also Hilbert space, with norm $||.||_{k}$."

My doubts:

a) What is $k$-linear symmetric continuous functions on $H$?
Precisely, what does symmetric mean? Is it $T(x,y,z)= T(z,y,x)$ for $x,y,z \in H$?

b) Which norm $||.||_{k}$ is this?

Thank you!

Answer 1

I just want to advertise this counterexample due to Robert Bonic, which I find much more appealing than Kupka's.

Consider $E=C^0([0,1])$ and $G:E\rightarrow E$ which sends $G(f)(x)=f(x)^3$. Then $G$ is smooth and $dG(f)(u)=3f^2 u$. Thus if $f$ has a zero, $f$ is a critical point for $G$. Now set $$ C=\{f\in E\,|f(x)<0,\quad f(y)>0,\quad \text{for some}\quad x,y\in[0,1]\} $$ Note that any $f\in C$ has a zero, and that $C$ is open. Moreover $G(C)= C$ hence the set of critical values of $G$ contains an open set.

Answer 2

I found the definition to $k$-linear symmetric, here Representation of symmetric functions.

And the norm $||.||_{k}$, http://www.math.uni-oldenburg.de/preprints/get/source/flo-gar.pdf page 2. Just change $k=2$ for $k$ arbitrary.