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Let $l_2^{+}$ be the Hilbert space of all square summable sequences $\{x_n\}, n \in \mathbb{N}$ under some definition of inner-product $\langle,\rangle_l$. Define $B[l_2^{+}]$ as the set of all bounded linear transformation from $l_2^{+}$ to $l_2^{+}$. Finally, let $A[l_2^{+}]$ be the set of all bounded transformations from $l_2^{+}$ to $l_2^{+}$. Note that $B[l_2^{+}] \subset A[l_2^{+}]$ and that $A[l_2^{+}]$ contains transformations that are not necessarily linear (the superposition property does not hold).

Define the inner product on $A[l_2^{+}]$:

$\langle T_1,T_2\rangle = \sup_{x \in l_2^{+} - \{0\}} \frac{\langle T_1x,T_2x \rangle_l}{\langle x,x \rangle_l}\text{, for all }T_1, T_2 \in A[l_2^{+}]$

Question: is $B[l_2^{+}]$ dense on $A[l_2^{+}]$?

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    Because of the sup, it is not linear in the first argument.2012-09-16

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