# Subspaces of vector space and their spans

$\def\sp{\operatorname{sp}}$ 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?

• Subspaces of V. @Zilliput – Nir Rauch Dec 20 '14 at 19:43
• Ah, ok. Then, by the definition of subspace, the span of a subspace is just that subspace. – Zilliput Dec 20 '14 at 19:44
• $T$ and $K$ are diffrent subspaces. Does it mean that #sp(K)=sp(T)$? – Nir Rauch Dec 20 '14 at 19:46 • If$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. – Michael Albanese Dec 20 '14 at 20:19