= Resolvent components in a closed unital subalgebra
{title2=$\rho_B(x)\subseteq\rho_A(x)$}
For a <closed> <unital> <subalgebra> $B$ of a <unital> <Banach algebra> $A$ and $x\in B$, the smaller-algebra <algebra resolvent set> is relatively <open> and <closed> in the larger one. Openness follows from the <Neumann series>; closedness follows from <continuity> of inversion in $A$ and <norm> closedness of $B$. Every larger-algebra resolvent component is therefore either retained whole or absorbed into the smaller-algebra <Banach algebra spectrum>. The unbounded component is always retained, since inverses at large modulus are norm-convergent series in $x$.
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