# Compact normed linear space

Does there exist a compact normed linear space?

All I can see is that if there is any it must be finite-dimensional. But does there exist any other than the trivial one i.e. $$(0)$$?

• Is the real line or the complex line compact? – kimchi lover Feb 16 at 18:12
• No @kimchilover.............. – Jave Feb 16 at 18:13
• There you are, then: if there is a non-zero element of your space, it contains a copy of the scalar multiples of that element. – kimchi lover Feb 16 at 18:14
• – Math1000 Feb 16 at 20:36