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In the general reflection convention, a root system in a finite-dimensional real inner product space $E$ is a finite spanning set $\Phi \subset E\setminus \{0\}$ such that every root reflectionpermutes $\Phi$. In particular $-\alpha \in \Phi$ whenever $\alpha \in \Phi$. A reduced root system additionally requires $\Phi \cap \mathbb{R}\alpha =\{\alpha ,-\alpha \}$. A crystallographic root system additionally requires all Cartan integers $2(\beta ,\alpha )/(\alpha ,\alpha )$ to be integers. Some authors include these extra conditions in the term root system; they are stated separately here.

For example, the A2 root system is $\{e_i-e_j:i\ne j\}$ in the plane $x_1+x_2+x_3=0$. Its six roots form a regular hexagon, and their reflections generate a Weyl group isomorphic to $S_3$. The roots of a complex semisimple Lie algebra form a reduced crystallographic root system: its root-space decomposition converts infinitesimal Lie brackets into this finite Euclidean geometry. The Dynkin diagram then records enough simple-root data to classify its simple components.^[ref]

The spectral theorem represents a normal operator by multiplication by its spectral parameter. In a finite-dimensional inner product space, this is diagonalization in an orthonormal basis; for a self-adjoint operator on a Hilbert space, it uses a projection-valued measure. Unbounded operators additionally require their natural square-integrability domain.

For a linearly independent ordered family in a real inner product space, require each unit vector $w_j$ to lie in the span of the first $j$ inputs and be orthogonal to all earlier inputs. That orthogonal part of the successive span is one-dimensional. Subtracting the earlier orthonormal components gives its nonzero generator $r_j$. Every possible unit choice is therefore $\pm r_j/\Vert r_j\Vert$. Since $⟨r_j,u_j⟩=\Vert r_j{\Vert }^2>0$, the positive normalization fixes the sign uniquely. This distinguishes uniqueness of the normalized ordered construction from the many possible orthonormal bases of the whole space.

A unitary operator $U:H\to K$ is a surjective linear isometry between complex inner product spaces. It preserves every inner product, not just the norm of a selected vector:Its adjoint operator is its inverse function, soFor an operator on one space, both identities have the same identity operator on the right. In finite-dimensional orthonormal bases, its matrix is a unitary matrix; the real counterpart is an orthogonal transformation.

Surjectivity matters in infinite dimension. An isometric embedding can satisfy $U^{\ast }U=I$ without satisfying $U{U}^{\ast }=I$: a proper image leaves some vectors outside its range. A unitary operator instead gives a reversible change of orthonormal coordinates, preserving all lengths and orthogonality.

A universe for spectra is a countably infinite-dimensional real inner product space $U$, used to index finite-dimensional suspension directions. A standard model is the algebraic direct sum $U={⨁}_{i\ge 1}\mathbb{R}{e}_i$ with an orthonormal basis, carrying its direct-limit topology. Its vectors have finite support; this universe is not the completed Hilbert space ${\ell }^2$. Indexed topological spectra assign spaces to finite-dimensional vector subspaces $V\subseteq U$ and use the corresponding orthogonal differences for structure maps.^[ref]

A family of indexing vector subspaces is cofinal when every finite-dimensional $W\subseteq U$ is contained in some member of the family. The flag $V_n=\mathrm{span}(e_1,\dots ,e_n)$ is cofinal: a finite basis of $W$ uses only finitely many of the $e_i$. In contrast, $W_n=\mathrm{span}(e_2,e_4,\dots ,e_{2n})$ has unbounded dimension but never contains $e_1$, so it is not cofinal. Cofinality ensures that restricting the index system does not discard any finite collection of suspension directions.