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$l_1$ is our (scalar) sequence space such that its norm $\Vert (a_n)\Vert=\Sigma_{n=1}^{\infty}|a_n|$ is finite. Let $(e_n)$ be the sequence of unit vectors in $l_1$.

Given $X$, a separable Banach space, and $(x_n)$, a sequence in $B_X$ that is dense in $B_X$. Define a linear operator $Q:l_1\rightarrow X$ by $$Q(e_n)=x_n,$$ then it says that in the light of $l_1$-norm property $Q$ is well-defined, bounded. I cannot put my finger on it. How can I show $Q$ is bounded?

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You have $$\| Q(x) \| \le \sum_{i=1}^\infty |x_n| \, \|Q(e_n)\| \le \sum_{i=1}^\infty |x_n| = \|x\|\ .$$

This shows that $Q$ is bounded.

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  • $\begingroup$ Thank you! I originally bounded each $a_n$ so I was stuck :( $\endgroup$ – CSH Jan 12 '17 at 8:49

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