Let $[a,b]$ be an interval and $X$ a Banach space (for starters). We know that continuous functions $f:[a,b]\to X$ are Riemann integrable. Suppose now that $X$ is a quasi-Banach space, that is, its norm satisfies $\|x+y\|\leq K (\|x\|+\|y\|)$ for all $x,y\in X$ and some $K\geq 1.$

I found that, in general, quasi-Banach spaces (or $p$-Banach spaces) do not have this nice integrability property. Someone needs a notion of analyticity, see Albiac–Ansorena, 2013 (DOI link).

However, I am mainly interested in the case where $X$ is the $\mathcal{L}^p(H)$ Schatten *-ideal on a separable Hilbert space $H$, where $p\in (0,1)$. Note that for $p\geq 1$ it is a Banach space (Bnach $*$-ideal). Some facts about this quasi-Banach space (case $p\in (0,1)$).

- Every $T\in \mathcal{L}^p(H)$ is compact
- For $T\in \mathcal{L}^p(H)$ the quasi-norm is $\|T\|_p= \| s_n(T)\|_{\ell_p}$, where $(s_n(T))_{n\geq 0}$ is the sequence of singular values in decreasing order, counting multiplicity.
- For $A,B\in \mathcal{B}(H)$ and $T\in \mathcal{L}^p(H)$ it holds that $\|ATB\|_p\leq \|A\|_{\mathrm{op}}\|T\|_p\|B\|_{\mathrm{op}}$
- Clearly, for $T\in \mathcal{L}^p(H)$ we have $\|T\|_{\mathrm{op}}\leq \|T\|_p$ since $s_0(T)=\|T\|_{\mathrm{op}}$.

**Question:** Is every continuous function $f:[a,b]\to \mathcal{L}^p(H)$ Riemann integrable ?

Thank you!