Alex, your idea of using a separation theorem can be turned into a full proof as follows:
As you said, AnotsubsetB iff there is a ainA and vector x such that supbinBlangleb,xrangle<langlea,xrangle. But then,
begin{align}
sup_{z in B + C} langle z, x rangle &= sup_{b in B, c in C} (langle b, x rangle + langle c, x rangle ) \\
&= sup_{b in B} langle b, x rangle + sup_{c in C} langle c, x rangle \\
&< langle a, x rangle + sup_{c in C} langle c, x rangle \\
&= sup_{z in a + C} langle z, x rangle \\
&leq sup_{z in B + C} langle z, x rangle,
end{align}
a contradiction.
No comments:
Post a Comment