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A Banach algebra is an associative algebra over $\mathbb{R}$ or $\mathbb{C}$ whose norm makes it a Banach space and satisfies submultiplicativity:An identity is an additional assumption; when present, its normalization $\Vert 1\Vert =1$ should be stated if needed. Submultiplicativity makes multiplication continuous, and completeness makes norm-controlled infinite algebraic operations possible.^[ref]

Examples include bounded linear operators on a Banach space with composition and the operator norm, and continuous complex-valued functions on a compact Hausdorff space with pointwise multiplication and the supremum norm. The former is generally noncommutative, whereas the latter is commutative. A closed subalgebra of a Banach algebra inherits both the product estimate and completeness.

For $\Vert a\Vert <1$, the powers satisfy ${\sum }_n\Vert a^n\Vert <\infty$, so the Neumann series constructs $(1-a{)}^{-1}$. This simple estimate connects analysis to invertibility and leads to the spectrum of an element and holomorphic functional calculus.

For a closed subalgebra of a Banach algebra $B\subseteq A$ with common identity, the algebra resolvent set ${\rho }_B(x)$ is clopen in ${\rho }_A(x)$. A point in ${\rho }_A(x)$ has a small connected disc on which membership of ${\rho }_B(x)$ is constant, so it cannot lie in $\partial {\sigma }_B(x)$. Also ${\sigma }_A(x)\subseteq {\sigma }_B(x)$; a point in the interior of the former is in the interior of the latter. Together these observations prove the boundary inclusion, without assuming a spectral-boundary theorem.