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Consider the linear space $c^{(3)}$ of all sequences $x = (x_n)_{n=1}^{\infty}$ such that $\{x_{3k+q} \}_{k=0}^{\infty}$ converges for $q = 0,1,2$. Find the dimension and a basis for $c^{(3)}/c_0$. Note that $c_0$ is the linear space of sequences that converge to $0$.

I think the dimension is $1$ using the reasoning in the answer to this question. We know that any integer can be represented as $3k, 3k+1$ or $3k+2$. So $c^{(3)}$ is equivalent to $c$ (e.g. all convergent sequences)? Is this correct?

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2 Answers 2

up vote 4 down vote accepted

Notice that $c^{(3)}\cong c\oplus c \oplus c$ where $c$ is the space of convergent sequences, while $c_0 \cong c_0 \oplus c_0 \oplus c_0$ (both isomorphisms are separating the sequence to its three sub sequences modulo 3).

Also, in the represetation above of $c^{(3)}$ and $c_0$ we get that each copy of $c_0$ is inside a copy of $c$ so $c^{(3)}/c_0 \cong (c/c_0) \oplus (c/c_0) \oplus (c/c_0)$

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Aren't the series $\{x_{3k} \}_{k=0}^{\infty}$, $\{x_{3k+1} \}_{k=0}^{\infty}$, and $\{x_{3k+2} \}_{k=0}^{\infty}$ completely unrelated to each other? They can each have a different limit.

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