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Find two linearly independent vectors $A,B ∈ $ $\ell^{\infty }$ s.t. $||\mathbf{A}||_{\infty}=||\mathbf{B}||_{\infty}=1$ and $||\mathbf{A+B}||_{\infty}=2$

What I know:

  • $\ell^{\infty}$ is a vector space where $\ell^{\infty}=[{\{A_n\}_{n=1}^\infty|\sup_{n\in \mathbb{N}} |A_n|< \infty}]$ equipped with norm $||{\{A_n\}_{n=1}^\infty}||=\sup_{n\in \mathbb{N}} |A_n|$.

  • For the triangle inequality, suppose that $\{A_n\},\{B_n\}\in \ell^{\infty}$. For each index $n$ we have $$|A_n+B_n|\leq |A_n|+|B_n|\leq \sup_{m}|A_m|+\sup_m|B_m|=||A||_{\infty}+||B||_{\infty}$$ Then taking the supremum over all $n$ $$ ||A+B||_{\infty}\leq ||A||_{\infty}+||B||_{\infty} $$

What I am confused on (linear independency and supremum): Do I assume that both vectors are already linearly independent, or do I need to show that there exist 2 scalars C1 and C2 that are zero s.t. no vector in the set can be represented as a linear combination of the remaining vectors in the set?

Does this mean that there is only one element in the norm after taking the supremum? $||\mathbf{A}||_{\infty}=1$

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  • 1
    $\begingroup$ Your starting point is wrong. You'll need to write down an example of A and B that satisfy all these properties, and there are many correct choices. The following observation might help you do this: prove that in any vector space $V$, a pair of vectors $x$ and $y$ is linearly dependent if and only if $x = cy$ for some scalar $c$. From here, to solve the exercise you posed, I recommend you start with $A = (1,0,0,...)$ and then try to find the correct $B$ using the above observation and the definition of the norm you supplied. $\endgroup$ – Jack Burkart Feb 1 '17 at 4:22
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Supplying an answer to follow up on the comment I posted above.

The sequences $A = (1,0,0,\dots)$ and $B = (1,1,0,\dots)$ satisfy all the conditions stated in the problem.

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