Embedding of Lp spaces

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.

• Unless I'm misunderstanding you, $L_p$ has a norm topology and so - I'm unlikely to make the right choice of terms here between basis and subbasis - is generated by open balls about each point. Oct 23, 2014 at 19:21
• Well, of course you're right. The trouble I'm facing is rather finding any properties of the open basis sets such as $B(f,a)=\{g \in L_p: (\int_{0}^{1}|f-g|^p\})^{1/p} < a$. I simply can't imagine a way to find $f \in L_p$ such that $B(f,a)\cap \text{im}(L_q) = \emptyset$ Oct 23, 2014 at 19:30

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.
• So you say then that $L_q$ is meager in $L_p$? May 10, 2016 at 4:11