L2 space is a Hilbert space ($L^2(X,\mathcal{F},\nu )$)

The space $L^2(X,\mathcal{F},\nu )$ consists of almost-everywhere equivalence classes of measurable functions with $\int |f{|}^{2}d\nu <\infty$. The inner productinduces its norm, and the Riesz-Fischer theorem makes it complete; hence it is a Hilbert space.

The exponentially weighted Hilbert space consists of measurable functions with ${\int }_{\mathbb{R}}|u(x){|}^2{e}^{2\gamma x}dx<\infty$. Multiplication by $e^{\gamma x}$ is a unitary map onto L2 space. Requiring unweighted square integrability as well produces an intersection that is incomplete under the weighted norm alone. For example, $1_{[-N,0]}$ is a weighted Cauchy sequence for $\gamma >0$, but its weighted limit $1_{(-\infty ,0]}$ is not in unweighted L2 space.

The linearization at a travelling wave $w_c$ is $A={\partial }_z^2+c{\partial }_z+k-2w_c$. Conjugation by $e^{\gamma z}$ changes this to ${\partial }_z^2+(c-2\gamma ){\partial }_z+{\gamma }^2-c\gamma +k-2w_c$. The choice $\gamma =c/2$ removes the first derivative and makes the operator self-adjoint. Its two limiting potentials are $-k-c^2/4$ and $k-c^2/4$, so essential-spectrum invariance under relatively compact perturbations gives ${\sigma }_{\mathrm{ess}}=(-\infty ,k-c^2/4]$. This lies strictly below zero for $c>2\sqrt{k}$ but touches zero at the critical speed. Stability of this essential spectrum alone is not a general nonlinear stability theorem.

For a probability measure $P$, the mean-zero L2 space isThe expectation functional is bounded: Cauchy-Schwarz inequality gives $|\int hdP|\le \Vert h{\Vert }_2$ because $P(\Omega )=1$. Its kernel is consequently a closed subspace of a Hilbert space. The inherited inner product isFor real functions the conjugate can be omitted; for complex functions the product $\int hkdP$ alone is not an inner product.

Every square-integrable function has the orthogonal decomposition $h=(h-Ph)+Ph$, with the first term in $L_0^2(P)$ and the second constant. The squared norm of the centered term is the variance in the real-valued case. This is the natural space for centered score functions and canonical gradients.

The L2 inner product is the inner product $⟨f,g⟩=\int f\overline{g}d\mu$ on L2 space. Its induced norm is the L2 norm, and the mass matrix represents it in a finite element basis.