Let us consider the Banach spaces $~~(L_p[0,1],\|.\|_p)~~$ and $~~(L_q[0,1],\|.\|_q)~~$ with $~~1 \le p<q\le \infty.~~$ We know that $~~L_q[0,1]~~$ is a proper subspace of $~~L_p[0,1].~$ The norms are defined as $$\|f\|_p=\left(\int_0^1|f(t)|^pdt\right)^{1/p}~~~\text{ and }~~~\|f\|_q=\left(\int_0^1|f(t)|^qdt\right)^{1/q}.$$ Now, I want to know that, is $~~L_q[0,1]~~$ is a closed proper subspace of $~~L_p[0,1]?~$ I want to find a sequence $~~f_n~~$ in $~~L_q[0,1]~~$ such that $~~\lim_{n \to \infty} f_n \notin L_q[0,1].~~$ Can you please give some time to solve this?
2026-05-14 05:00:06.1778734806
$L_q[0,1]~~$ is a closed proper subspace of $~~L_p[0,1]$
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Actually, $L_q[0,1]$ contains the linear combinations of indicator functions, hence the closure of $L_q[0,1]$ in $L_p[0,1]$ for the $\lVert \cdot\rVert_p$ norm contains the closure of he linear combinations of indicator functions in $L_p[0,1]$ for the $\lVert \cdot\rVert_p$ norm, which is $L_p[0,1]$.
In other words, $L_q[0,1]$ is a dense subspace of $L_p[0,1]$ for the $\lVert \cdot\rVert_p$ norm.