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If an array is antisymmetric in every orthonormal basis and its contractions with every antisymmetric second-rank tensor are invariant scalars, it obeys the Cartesian second-rank tensor transformation law. The difference $D=R^{T}A\prime R-A$ is antisymmetric and orthogonal to every antisymmetric test matrix. Choosing the test matrix equal to $D$ gives $D:D=0$, hence $D=0$. The Frobenius inner product is nondegenerate on the antisymmetric subspace.

An antiunitary operator between complex Hilbert spaces is a surjective antilinear map satisfying the displayed inner product identity. It preserves norms but conjugates scalars. In a chosen orthonormal basis it is $UK$, with $U$ a unitary operator and $K$ coefficientwise complex conjugation. Conversely any such $UK$ is antiunitary. Antilinearity by itself does not preserve norms.

The best N-term approximation in an orthonormal basis retains $N$ coefficients of greatest absolute value. It minimizes the squared Hilbert space error over all index sets of size at most $N$. The selection makes the approximation nonlinear in the data, even though coefficient extraction is linear.

For a complete orthonormal basis of $L^2({\mathbb{R}}_{+})$, the Parseval identity for a Hilbertian basis identifies the sum of the feature products with $⟨1_{[0,s]},1_{[0,t]}⟩=\min (s,t)$. The Cauchy-Schwarz inequality gives absolute convergence.

For a centered Gaussian measure on a real Hilbert space with injective covariance operator of a Gaussian measure $\Sigma$, its Cameron-Martin space is $E_{\mu }=\mathrm{Ran}{\Sigma }^{1/2}$, with inner product $⟨h,g{⟩}_{E_{\mu }}=⟨{\Sigma }^{-1/2}h,{\Sigma }^{-1/2}g⟩$. In an orthonormal basis satisfying $\Sigma e_j={\lambda }_j{e}_j$, it consists of exactly the vectors $h$ for which ${\sum }_j{h}_j^2/{\lambda }_j<\infty$. It is complete in this stronger norm even if its range is not closed in the ambient norm. Its importance is that it describes precisely the translations that preserve the null sets of the Gaussian measure, as in the Cameron-Martin theorem for a Gaussian measure. For generalized Gaussian white noise over $L^2$, the corresponding Cameron-Martin space is L2 space itself.

A Cartesian second-rank tensor has component matrix $T$ in each orthonormal basis, with transformation $T\prime =RT{R}^T$ under an orthogonal component change $R$. Both indices transform; this is why assigning arbitrary matrices independently in different bases does not define a tensor. The determinant, trace, and Frobenius inner product with another such tensor are unchanged by these basis transformations.

If $V$ is a subspace of a space with an inner product and $P_V$ is its orthogonal projection, the compression of an operator $A$ to $V$ isIf the columns of $Z$ are an orthonormal basis of $V$, its matrix in that basis is $Z^{\ast }AZ$.

For any orthonormal basis and Hermitian operator, ${\sum }_i|⟨{\psi }_i|X|{\psi }_i⟩|\le \Vert X{\Vert }_1$. In the trace-norm variational principle for Hermitian operators, choose the diagonal $T$ whose entries are the signs of these diagonal entries. This bounds the total distinguishability visible in one fixed measurement basis.

A smooth uniformly elliptic differential expression with symmetric Dirichlet bilinear form has an unbounded self-adjoint operator realization $A_D$ on $L^2(U)$, whose domain is $H^2(U)\cap H_0^1(U)$ on a smooth bounded domain. If the form is strictly positive, the Lax-Milgram theorem and elliptic regularity make $A_D^{-1}$ a bounded map from $L^2$ to $H^2\cap H_0^1$. Its action on $L^2$ is compact by the Rellich-Kondrachov compactness theorem and self-adjoint by symmetry of the form. The spectral theorem for compact Hermitian operators therefore gives an orthonormal basis of smooth Dirichlet eigenfunctions with positive eigenvalues tending to infinity.

If a measure $\rho$ has finite Green energy, thenconverges in $L^2$ for any orthonormal basis $(e_n)$ of $H_0^1(D)$ and independent standard normal coefficients ${\xi }_n$. The limit is centered Gaussian with variance equal to the Green energy of $\rho$ and does not depend on the chosen basis.

The functions $\sqrt{2/\pi }\sin (nx)$, $n\ge 1$, form a complete orthonormal basis of $L^2(0,\pi )$. An odd extension to $(-\pi ,\pi )$ reduces completeness to that of the Fourier series. Equality of all sine coefficients therefore implies equality almost everywhere; if both functions are continuous, it implies equality everywhere.

For $f:\{-1,1{\}}^n\to \mathbb{R}$, the characters ${\chi }_S(x)={\prod }_{i\in S}{x}_i$ form an orthonormal basis, and the Fourier-Walsh expansion is

For the sum-zero standard representation of the symmetric group, the vectors $v_j=e_1+\cdots +e_j-j{e}_{j+1}$ form a Gelfand–Tsetlin basis. The Young–Jucys–Murphy element $X_i$ has eigenvalue $i-1$ for $i\le j$, $-1$ for $i=j+1$, and $i-2$ for $i>j+1$. Normalizing by $\sqrt{j(j+1)}$ gives an orthonormal basis corresponding to tableaux of shape $(n-1,1)$ whose second-row entry is $j+1$.

For a geodesic $\gamma$ with initial velocity $v$, $(f\circ \gamma {)}^{\prime \prime }(0)=\mathrm{Hess}f(v,v)$, because the covariant acceleration vanishes. Summing over an orthonormal basis of the tangent space gives $\Delta f(p)=-{\sum }_i(f\circ {\gamma }_i{)}^{\prime \prime }(0)$ for the nonnegative Laplace-Beltrami operator.

For an orthonormal basis $(e_n)$ of $H_0^1(D)$, the zero-Dirichlet Green function of the Laplacian is the weak kernel represented byConsequently, whenever the terms are defined and the right side is finite,

At level $J$, a Haar scaling function is the normalized indicator function $2^{J/2}{1}_{I_{J,k}}$ of a dyadic interval. The functions at that level form an orthonormal basis of the cellwise-constant subspace. Their normalization is what turns white-noise coefficient variances into $1/n$ and cellwise variances into $2^J/n$.

On level-one cusp forms, determinant-normalized Hecke operators satisfy $⟨T_{n}f,g⟩=⟨f,T_{n}g⟩$. Unfolding their finite correspondences transfers a matrix to its inverse; its adjugate has the same determinant and reverses the correspondence. Since these operators commute, the cusp space has a common orthonormal basis of Hecke eigenforms, with real eigenvalues.

The Hilbert-Schmidt norm of a Hilbert-Schmidt operator is the square root of the displayed sum. The value is independent of the orthonormal basis, by the Parseval identity. For a finite matrix, it is the Frobenius norm.

A bounded linear operator on a Hilbert space is Hilbert-Schmidt when ${\sum }_n\Vert T{e}_n{\Vert }^2<\infty$ for one, equivalently every, orthonormal basis. Every Hilbert-Schmidt operator is a compact operator.

For the Riemannian metric $d{r}^2+f(r{)}^{2}d{\varphi }^2$, where $f>0$ and $\varphi$ has period $2\pi$, parallel transport around $r=r_0$ acts on the orthonormal basis components of a tangent vector by a rotation through $-2\pi f^{\prime }(r_0)$. Indeed, those components obey $u^{\prime }=f^{\prime }(r_0)v$ and $v^{\prime }=-f^{\prime }(r_0)u$ as functions of $\varphi$. Thus all tangent vectors return to themselves exactly when $f^{\prime }(r_0)\in \mathbb{Z}$. The integer criterion concerns holonomy and is stronger than local flatness.