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Ady, I don't have an answer to the new version of your question but let me make some remarks which might be useful.

The new version is about non-linear real-valued continuous functions on $\ell_\infty(\Gamma)$ where $\Gamma$ has the cardinality of the continuum. This can be slightly generalized as follows:

Let $\kappa$ be an infinite cardinal and set $K$ to be the closed unit ball of $\ell_\infty(\kappa)$. Let $f:K\to\mathbb{R}$ be a continuous map. Does there exist an infinite-dimensional subspace $E$ of $\ell_\infty(\kappa)$ such that $f(K\cap E)$ is bounded?

If $\kappa=\aleph_0$, then a counterexample can be constructed.

On the other hand, if $\kappa$ is a measurable cardinal, then there exists a subspace $E$ of $\ell_\infty(\kappa)$ which is isomorphic to $c_0(\kappa)$ and such that $f(K\cap E)$ is bounded. The argument goes back to Ketonen. Let $FIN(\kappa)$ be the set of all non-empty finite subsets of $kappa$ and define a coloring $c:FIN(\kappa)\to\mathbb{N}$ as follows. Let $c(F)$ be $n$ if $n$ is the least integer $m$ such that

$\max{ |f(x)|: x\in span{e_t: t\in F} and x\in K } \leq m$

where $e_t$ is the dirac function at $t$. Notice that $c$ is well-defined. There exist $n_0\in\mathbb{N}$ and a subset $A$ of $\kappa$ with $|A|=\kappa$ and such that $c$ is constant on $FIN(A)$ and equal to $n_0$. If we set $E$ to be the closed linear span of ${e_t: t\in A}$, then $E$ is isomorphic to $c_0(\kappa)$ and $F(K\cap E)$ is in the interval $[-n_0, n_0]$.

Concerning the continuum: it might be that there are set-theoretic issues. Firstly, let me recall that it is consistent that the the continuum is real-valued measurable (R. M. Solovay). On the other hand, if CH holds, then there is heavy (and quite advanced) machinery for killing" various Ramsey properties on $\omega_1$ (largely due to S. Todorcevic).

A quick remark: there exists a non-linear continuous map $f:K\to\mathbb{R}$, where $K$ is the closed unit ball of $c_0(\kappa)$ and $\kappa$ is the continuum, such that for every infinite-dimensional subspace $E$ of $c_0(\Gamma)$ c_0(\kappa)$the set$f(K\cap E)$is unbounded. 4 added 291 characters in body Ady, I don't have an answer to the new version of your question but let me make some remarks which might be useful. The new version is about non-linear real-valued continuous functions on$\ell_\infty(\Gamma)$where$\Gamma$has the cardinality of the continuum. This can be slightly generalized as follows: Let$\kappa$be an infinite cardinal and set$K$to be the closed unit ball of$\ell_\infty(\kappa)$. Let$f:K\to\mathbb{R}$be a continuous map. Does there exist an infinite-dimensional subspace$E$of$\ell_\infty(\kappa)$such that$f(K\cap E)$is bounded? If$\kappa=\aleph_0$, then a counterexample can be constructed. On the other hand, if$\kappa$is a measurable cardinal, then there exists a subspace$E$of$\ell_\infty(\kappa)$which is isomorphic to$c_0(\kappa)$and such that$f(K\cap E)$is bounded. The argument goes back to Ketonen. Let$FIN(\kappa)$be the set of all non-empty finite subsets of$kappa$and define a coloring$c:FIN(\kappa)\to\mathbb{N}$as follows. Let$c(F)$be$n$if$n$is the least integer$m$such that$ \max{ |f(x)|: x\in span{e_t: t\in F} and x\in K } \leq m $where$e_t$is the dirac function at$t$. Notice that$c$is well-defined. There exist$n_0\in\mathbb{N}$and a subset$A$of$\kappa$with$|A|=\kappa$and such that$c$is constant on$FIN(A)$and equal to$n_0$. If we set$E$to be the closed linear span of${e_t: t\in A}$, then$E$is isomorphic to$c_0(\kappa)$and$F(K\cap E)$is in the interval$[-n_0, n_0]$. Concerning the continuum: it might be that there are set-theoretic issues. Firstly, let me recall that it is consistent that the the continuum is real-valued measurable (R. M. Solovay). On the other hand, if CH holds, then there is heavy (and quite advanced) machinery for killing" various Ramsey properties on$\omega_1$(largely due to S. Todorcevic). A quick remark: there exists a non-linear continuous map$f:K\to\mathbb{R}$, where$K$is the closed unit ball of$c_0(\kappa)$and$\kappa$is the continuum, such that for every infinite-dimensional subspace$E$of$c_0(\Gamma)$the set$f(K\cap E)$is unbounded. 3 added 183 characters in body Ady, I don't have an answer to the new version of your question but let me make some remarks which might be useful. The new version is about non-linear real-valued continuous functions on$\ell_\infty(\Gamma)$where$\Gamma$has the cardinality of the continuum. This can be slightly generalized as follows: Let$\kappa$be an infinite cardinal and set$K$to be the closed unit ball of$\ell_\infty(\kappa)$. Let$f:K\to\mathbb{R}$be a continuous map. Does there exist an infinite-dimensional subspace$E$of$\ell_\infty(\kappa)$such that$f(K\cap E)$is bounded? If$\kappa=\aleph_0$, then a counterexample can be constructed. On the other hand, if$\kappa$is a measurable cardinal, then there exists a subspace$E$of$\ell_\infty(\kappa)$which is isomorphic to$c_0(\kappa)$and such that$f(K\cap E)$is bounded. The argument goes back to Ketonen. Let$FIN(\kappa)$be the set of all non-empty finite subsets of$kappa$and define a coloring$c:FIN(\kappa)\to\mathbb{N}$as follows. Let$c(F)$be$n$if$n$is the least integer$m$such that$ \max{ |f(x)|: x\in span{e_t: t\in F} and x\in K } \leq m $where$e_t$is the dirac function at$t$. Notice that$c$is well-defined. There exist$n_0\in\mathbb{N}$and a subset$A$of$\kappa$with$|A|=\kappa$and such that$c$is constant on$FIN(A)$and equal to$n_0$. If we set$E$to be the closed linear span of${e_t: t\in A}$, then$E$is isomorphic to$c_0(\kappa)$and$F(K\cap E)$is in the interval$[-n_0, n_0]$. Concerning the continuum: it might be that there are set-theoretic issues. Let Firstly, let me recall that it is consistent that the the continuum is real-valued measurable (R. M. Solovay). On the other hand, if CH holds, then there is heavy (and quite advanced) machinery for killing" various Ramsey properties on$\omega_1\$ (largely due to S. Todorcevic).

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