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An interval-adapted wavelet basis is an orthonormal basis of $L^2[0,1]$ made of a finite coarse scaling function family and resolution-indexed wavelets. A localized construction modifies only a bounded number of functions near each endpoint at every level. Boundary modification must preserve nested refinement spaces and the required vanishing moments; simple restriction of a whole-line basis does not do so.

A linear map $V:\mathcal{H}\to \mathcal{K}$ is an isometry when it preserves inner products, equivalently $V^{†}V=I$. It need not be surjective. It maps an orthonormal basis to an orthonormal family, and $\rho \mapsto V\rho V^{†}$ preserves the nonzero eigenvalues and the Von Neumann entropy of a density operator.

A matrix coefficient of a finite-dimensional group representation is a scalar function obtained by applying a linear functional to the translate of a fixed vector. In a chosen basis these functions include $x\mapsto \rho (x{)}_{ij}$. For a finite group, the Schur orthogonality relations make the scaled coefficients of all inequivalent unitary irreducible representations an orthonormal basis of its scalar functions.

The maximally mixed density operator on an $N$-dimensional Hilbert space assigns equal weights to every vector of any orthonormal basis. It is invariant under every unitary operator. Its purity is $\mathrm{Tr}({\rho }_{\ast }^2)=1/N$, the minimum possible because the nonnegative eigenvalues sum to one and obey the Cauchy-Schwarz inequality. For $N>1$ it is a mixed state; for $N=1$ it is the sole pure state. A rank-$k$ projector has probability $k/N$ in this state.

A measurement channel writes a POVM's outcome into an orthogonal classical register:For any input orthonormal basis $|j⟩$, the Kraus operators $K_{a,j}=|a⟩⟨j|\sqrt{E_a}$ realize this map and satisfy ${\sum }_{a,j}{K}_{a,j}^{†}{K}_{a,j}=I$. It is therefore a trace-preserving completely positive map.

An N-term approximation in an orthonormal basis uses at most $N$ basis functions. For a chosen index set $\Lambda$, its optimal coefficients are the inner products with those functions, and its squared error is the sum of the omitted squared coefficients by the Parseval identity.

In a right-handed reference frame, the rotation matrix from an orbital plane frame whose first axis points along the ascending node is $R_z(\Omega )R_x(I)$. Its columns are the orbital frame's orthonormal basis expressed in reference coordinates. The unit vector normal is $(\sin \Omega \sin I,-\cos \Omega \sin I,\cos I)$; its dot product with another such normal gives the cosine of their mutual inclination.

An orthogonal basis of an inner-product vector space consists of nonzero mutually orthogonal vectors that span the space. The orthogonal projection of $v$ onto a finite-dimensional span is ${\sum }_i⟨v,u_i⟩u_i/⟨u_i,u_i⟩$. Dividing each vector by its norm produces an orthonormal basis. For centered random variables, the inner product is covariance; an orthogonal basis therefore consists of uncorrelated variables, without requiring unit variance.

A real symmetric matrix has real eigenvalues, orthogonal eigenspaces for distinct eigenvalues, and an orthonormal basis of eigenvectors. From diagonalizability, apply the Gram-Schmidt process within each eigenspace; all its linear combinations stay in that eigenspace. This gives $A=Q\Lambda Q^T$ with an orthogonal matrix $Q$ and real diagonal $\Lambda$.

An orthonormal eigenbasis of a linear operator is an orthonormal basis consisting of eigenvectors.