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The question is as follows: I want to find an isometric embedding between the space of all real bounded sequences $l^\infty$ and a subspace of continuous functions $C(0,1]$.

Here are my attempts. In fact, I can find a one-one correspondence between $l^\infty$ and a subspace of $C(0,1]$ by $$ (a_n)_{n\geq 1}\mapsto \sum\frac{a_n}{2^n}x^n $$ But obviously it is not an isometric embedding. So can any one help me with this problem? Thanks.

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  • $\begingroup$ Just an idea: Send $a_n$ to a function with $f(n) = a_n$, and to make it continuous you can have it interpolate linearly between $f(n)$ and $f(n+1)$. $\endgroup$ Sep 9, 2022 at 13:11

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Given $a$, define the continuous function by $f(\frac1n) = a_n$ such that it is affine on the intervals $[\frac1{n+1},\frac1n]$. Such piecewise affine and continuous functions also form a subspace.

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