subgeometries of ${\mathrm{PG}}(V,{{\mathbb F}}_{q^n})$. If $h=r-1$ and $\dim_{{{\mathbb F}}_q}(U)=n$, then $U$ is $h$-scattered exactly when $U$ defines a scattered ${{\mathbb F}}_q$-linear set with respect to the hyperplanes, introduced in [@ShVdV Definition 14]; see also [@Lunardon2017].

Theorem \[th:bound\] bounds the dimension of a $h$-scattered subspace.

\[th:bound\][[@CsMPZ Theorem 2.3]]{} If $U$ is a $h$-scattered ${{\mathbb F}}_q$-subspace of dimension $k$ in $V=V(r,q^n)$, then one of the following holds:

-   $k=r$ and $U$ defines a subgeometry ${\mathrm{PG}}(r-1,q)$ of ${\mathrm{PG}}(V,{{\mathbb F}}_{q^n})$;

-   $k\leq\frac{rn}{h+1}$.

A $h$-scattered ${{\mathbb F}}_q$-subspace of highest possible dimension is said to be a [*maximum $h$-scattered*]{} ${{\mathbb F}}_q$-subspace. Theorem \[th:inter\] bounds the dimension of the intersection between a $h$-scattered subspace of dimension $\frac{rn}{h+1}$ and an ${\mathbb{F}_{q^n}}$-subspace of codimension $1$.

[[@CsMPZ Theorem 2.8]]{} \[th:inter\] If $U$ is an $\frac{rn}{h+1}$-dimensional $h$-scattered ${{\mathbb F}}_q$-subspace of a vector space $V=V(r,q^n)$, then for any $(r-1)$-dimensional ${{\mathbb F}}_{q^n}$-subspace $H$ of $V$ we have $$\frac{rn}{h+1}-n\leq \dim_{{{\mathbb F}}_q}(U \cap H) \leq \frac{rn}{h+1}-n+h.$$

Constructions of $h$-scattered ${{\mathbb F}}_q$-subspaces have been given in [@CsMPZ] and also in [@NPZZ]. A generalization of $h$-scattered subspaces has been recently introduced in [@BCsMT].

Two dualities for $\mathbb{F}_q$-subspaces {#sec:dualities}
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In this paper we need both ordinary and Delsarte dualities.

### Ordinary duality {#sec:classicalduality}

Let $\sigma \colon V\times 