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An integral operator maps a function $f$ to a function of the formA bounded continuous kernel on a compact domain defines a bounded operator in the uniform norm.

A topological vector space is a vector space $E$ over $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$, equipped with a topology such that vector addition and scalar multiplication are jointly continuous:The domains carry the product topology, and the scalar field carries its usual topology. Joint continuity permits both inputs to vary at once. Here the definition does not impose the Hausdorff condition; we state it separately when needed. ^[ref].

Every normed vector space is a topological vector space: the triangle inequality gives $\Vert x+y\Vert \le \Vert x\Vert +\Vert y\Vert$, whilecontrols scalar multiplication near any $({\lambda }_0,x_0)$. More generally, a seminorm topology supplies many measurements of smallness, rather than one norm. Translation by any vector is a homeomorphism, so a neighbourhood basis at zero determines all neighbourhoods by translation.

The compatibility matters: on a nonzero real vector space, the discrete topology makes vector addition continuous, but not scalar multiplication. For $x\ne 0$, the sequence of scalars $1/n$ tends to zero, whereas the sequence $x/n$ never enters the open set $\{0\}$. Thus choosing a topology arbitrarily need not respect the linear algebra.

This framework lets functional analysis choose the appropriate notion of limit of a sequence. Norm topology measures one global error, weak topology tests linear functionals, and the Fréchet space of smooth functions controls every order of derivative. In each case the operations forming linear combinations are continuous, making approximation compatible with the underlying equations.

A linear operator $T:X\to Y$ is weakly compact when $T(B_X)$ is relatively weakly compact in $Y$.

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.

The weak topology $\sigma (X,X^{\ast })$ of a normed vector space is the coarsest topology making every continuous linear functional $f\in X^{\ast }$ continuous. A basic neighborhood of $x$ imposes finitely many tests $|f_j(y-x)|<ϵ$. A net converges weakly exactly when all these scalar evaluations converge.

Norm convergence implies weak convergence, since $|f(x_n-x)|\le \Vert f\Vert \Vert x_n-x\Vert$. The converse can fail: the standard unit vectors in ${\ell }^2$ converge weakly to zero, because every square-summable sequence has coordinates tending to zero, but their norms stay one. Weak topologies are useful because bounded families can have compactness properties unavailable in the norm topology, particularly in reflexive spaces. The tradeoff is that weak limits do not preserve every nonlinear operation.

A Banach space $X$ is reflexive when its canonical embedding $J_X:X\to X^{\ast \ast }$ is surjective. Equivalently, its closed unit ball is weakly compact.

A Schauder basis of a Banach space $X$ is a sequence $(e_n)$ such that every $x\in X$ has a unique norm-convergent expansion $x={\sum }_{n=1}^{\infty }{a}_n{e}_n$.

Conformal time satisfies $d\tau =dt/a(t)$ and factors the FLRW metric into $a^2(\tau )$ times a static conformal metric.

Proper distance is the spatial metric length measured along a fixed-cosmic-time slice.

The cases $k>0$, $k=0$, and $k<0$ give spherical, Euclidean, and hyperbolic constant-curvature spatial slices.

In Cartesian comoving coordinates and signature $(-,+,+,+)$, a spatially flat FLRW universe has

If the present radiation and cosmological-constant densities are respectively $\beta$ and $1$ times the present critical density, while $a_0=1$, positive curvature givesTurning points are therefore the positive roots of $x^2-\beta x+\beta$ with $x=a^2$.

For energy density $\rho$, pressure $P$, and cosmological constant $\Lambda$,

Combining the perfect-fluid continuity equation withshows thatis constant in time. This constant is the spatial-curvature integration constant in the first Friedmann equation.

A self-similar object is assembled from smaller copies of itself, possibly after scaling or other transformations.

With the standard Fourier convention, Plancherel theorem applied to $1_{[-1,1]}$ gives

With the sign conventionthe Hilbert transform is a singular integral operator. For a Schwartz function, the Cauchy principal value removes equal small intervals on both sides of $y=x$. Subtracting $f(x)$ locally makes the remaining quotient bounded there, while rapid decay controls infinity. Reversing the denominator reverses the operator sign.

For the angular-frequency Fourier transform, its Fourier multiplier is $-i\mathrm{sgn}\xi$, as derived under Hilbert-transform Fourier multiplier. Plancherel theorem therefore extends it to an isometry on $L^2(\mathbb{R})$, and $H^2=-I$ on that space because the squared multiplier is minus one almost everywhere. This frequency-dependent phase shift connects real and imaginary boundary parts of suitable holomorphic functions; Hilbert transform of the sinc function supplies an explicit example.

The transform need not preserve boundedness. For the indicator function of $[-1,1]$, direct integration givesIt diverges logarithmically near the endpoints and has tail $2/(\pi x)+O(x^{-3})$. This example also explains why a compactly supported input can produce a long-range response. The finite Hilbert transform instead restricts the integration interval and has its own endpoint and null-space issues.

For the angular-frequency convention $\hat{f}(\xi )=\int e^{-ix\cdot \xi }f(x)dx$, the inverse-distance tempered distribution on ${\mathbb{R}}^3$ has the exact transformBoth sides are locally integrable in three dimensions and have at most polynomial growth, so they define distributions even though the original function is not integrable at infinity.

One direct proof introduces $f_{ϵ}(x)=e^{-ϵ|x|}/|x|$, with $ϵ>0$. Angular integration of the radial Fourier transform gives, for $s=|\xi |>0$,The damped sine integral follows by taking the imaginary part of $(ϵ-is{)}^{-1}$. Letting $ϵ\downarrow 0$ against a Schwartz function is justified by the locally integrable majorants $|x{|}^{-1}$ and $|\xi {|}^{-2}$ together with rapid decay. This proves the distributional identity with its constant.

The result agrees with $-\Delta (1/|x|)=4\pi {\delta }_0$: the Fourier transform of a derivative multiplies the transformed potential by $|\xi {|}^2$. Changing to a cycles-per-unit convention changes the numerical coefficient, so the transform convention is part of the statement.

If $f_{\lambda }(x)=f(\lambda x)$ on ${\mathbb{R}}^n$ with $\lambda >0$, thenand consequently

On $(-\pi ,\pi )$,Parseval's identity and direct integration of the square give