0
$\begingroup$

I tried to prove the above statement showing that the boundary of each proper set is non-empty. But i could not did it then i tried to construct a line joining each two points in the set Banach space, and show that each such line is connected, to show that each line is connected i tried to prove that each proper set is non-empty using supremum and infimum, i could do this, but whenever the supremum and the infimum of the proper set is one of the points i could not show that the boundary of the proper set is not empty.

How it may be proven?

ps: The space does not need to be complete, furthermore we may assume the scalar field to be either the real or complex numbers.

  • 0
    I suppose your scalar field is $\mathbb{C}$ or $\mathbb{R}$? In any case, the map $t \mapsto t\cdot x$ is continuous for every $x$ in the Banach (or more generally topological vector) space. What do you know about continuous maps and connectedness? – 2017-01-31
  • 0
    yes Fischer you is right, i did not told anything about the scalar field because the similar algebraic properties, but now is more clear i finished the proof using the mapping you suggested, once the image is connected i could finish the proof )) thank you very much – 2017-01-31
  • 0
    should i edit the question? – 2017-01-31
  • 1
    It doesn't matter whether one looks at real or complex vector spaces, but if one looks at Banach spaces over the $p$-adic numbers, or over $\mathbb{Q}$, or some other disconnected field, that matters. But those aren't very often used, so it's perhaps not necessary to edit. However, it wouldn't harm to clarify that the scalar field should be one of $\mathbb{R}$ or $\mathbb{C}$. – 2017-01-31

0 Answers 0