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$\def\sp{\operatorname{sp}}$ Given 2 subspaces of V, and $T\subseteq \sp(K)$ and $K\subseteq \sp(T)$ then $\sp(K)=\sp(T)$?

If $ T\subseteq \sp(T)\subseteq \sp(K)$ and $K\subseteq \sp(K)\subseteq \sp(T)$ then it is true. But in general this statement is wrong.

But I think I am missing something. Any ideas?

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  • $\begingroup$ Subspaces of V. @Zilliput $\endgroup$ – Nir Rauch Dec 20 '14 at 19:43
  • $\begingroup$ Ah, ok. Then, by the definition of subspace, the span of a subspace is just that subspace. $\endgroup$ – Zilliput Dec 20 '14 at 19:44
  • $\begingroup$ $T$ and $K$ are diffrent subspaces. Does it mean that #sp(K)=sp(T)$? $\endgroup$ – Nir Rauch Dec 20 '14 at 19:46
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    $\begingroup$ If $T$ and $K$ are subspaces and $\sp$ is span, then $\sp(T) = T$ and $\sp(K) = K$. The hypotheses $T \subseteq \sp(K)$ and $K \subseteq \sp(T)$ are therefore equivalent to $\sp(T) \subseteq \sp(K)$ and $\sp(K) \subseteq \sp(T)$; it follows immediately that $\sp(K)= \sp(T)$. A more interesting question is whether $T \subseteq \operatorname{K}$ and $K \subseteq \sp(T)$ imply $\sp(K) = \sp(T)$ in the case where $T$ and $K$ are just sets of vectors. $\endgroup$ – Michael Albanese Dec 20 '14 at 20:19

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