I've managed to prove that for $ 1\leq p < q \leq +\infty $ we have an inclusion (embedding) $ L_q([0,1],\lambda) \rightarrow L_p([0,1], \lambda)
~~ (\lambda $ being Lebesgue measure). The trouble I'm facing now is:
a) is this embedding continuous? I can imagine why a preimage of an open subset in $ L_p $ should be open in $ L_q $, but I can't really prove that.
b) Is the image of this embedding a dense set? A closed set?
I'm having trouble understanding what open sets in $ L_p $ might be.
Best Answer
We have the relationship $\lVert f\rVert_p\leqslant \lVert f\rVert_q$ for each function $f$. Since the embedding is linear, it follows that it is continuous.
The image contains $\mathbb L^\infty$, which is dense. It can be written as a countable union of closed sets with empty interior.