I recently started a course in functional analysis and the teacher assumes we are familiar with some notation and concepts, but I feel completely lost when it comes down to the L spaces.
For example let me cite a part of a homework assignment he gave us:
Problem 1. In this problem we consider two norms on L^1(0, 1).
(a) Show that L^2(0, 1) is contained in L^1(0, 1) as a dense subspace.
Here what I think $L^1(0,1)$ is: norm space of all measurable functions (from the interval $(0,1)$) equipped with the $1$ norm. Similarly, $L^2$ is the same thing but equipped with the $2$ norm.
Here is where confusion comes in: how can one norm space be contained within another?
Or is $L^1$ the set of all measurable function with respect to the $1$ norm? Then the question would make sense to me, but I'm unsure of what $L^1$ looks like.
Could someone clarify this for me by giving me proper definitions (and some idea on what I'm dealing with) of L spaces?
Let $1\leqslant p<\infty$, $L^p([0,1])$ is the set of all mesurable functions on $[0,1]$ such that $f^p$ is integrable on $[0,1]$ modulus the equality almost everywhere. In other words, one has: $$L^p([0,1]):=\left\{f:[0,1]\rightarrow[0,1]\textrm{ mesurable s.t. }\int_0^1|f|^p<\infty\right\}/\sim.$$ The norm on $L^p([0,1])$ is the following: $$\|f\|_{L^p([0,1])}:=\left(\int_0^1|f|^p\right)^{1/p}.$$
Remark. $L^p(X,\mathbb{C})$ makes sense whenever $X$ is a measure space, the definition remains the same, just replace $[0,1]$ by $X$.