I am trying to prove that a locally closed subset of an l-space is also an l-space.
An l-space is defined as a Hausdorff, locally compact, zero-dimensional topological space.
I am having difficulty in proving the locally compact part. I know it is true that a closed subset of a locally compact space is locally compact. Is it also true for a locally closed subset? Thanks!
As we are working in Hausdorff spaces, all usual definitions of local compactness are equivalent. (i.e. each point having a compact neighbourhood, or having a base of compact neighbourhoods, or having a local base with compact closures etc.).
All subspaces of a zero-dimensional (in the sense of having a clopen base) spaces are trivially zero-dimensional again, so we need not bother about that aspect of what the OP calls $l$-spaces. Likewise for Hausdorffness.
A locally closed subset $A$ of $X$ is by definition a subset of the form $A = O \cap C$ where $O$ is open in $X$ and $C$ is closed in $X$.
$C$ in the subspace topology is locally compact (as a closed subspace) and $A = O \cap C$ is thus an open subspace of a locally compact subspace $C$) and thus in turn locally compact in its subspace topology too.