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The weak topology $\sigma (X,X^{\ast })$ on a normed space $X$ is the coarsest topology making every member of $X^{\ast }$ continuous. A net converges weakly exactly when every bounded linear functional converges on it.

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 one common sign convention,Reversing the denominator reverses the sign.

As a tempered distribution on ${\mathbb{R}}^3$,for a nonzero convention-dependent constant $C$. One proof writesand applies the Fourier transform of a Gaussian before substituting $u=|\xi {|}^2/(4t)$.

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

A Fourier cosine series has the formIt represents the even function obtained by reflecting its data across the origin.

For an integrable function of period $L>0$, its complex Fourier coefficients measure its inner product with each complex exponential mode:For a real function of period $2\pi$, the real coefficient convention of Fourier series gives ${\hat{f}}_0=a_0/2$ and ${\hat{f}}_n=(a_n-i{b}_n)/2$ for $n\ge 1$, with ${\hat{f}}_{-n}=\overline{{\hat{f}}_n}$. For $f(x)=1/2+\cos x+\frac{1}{2}\sin (2x)$, the only nonzero complex Fourier coefficients are ${\hat{f}}_0=1/2$, ${\hat{f}}_{\pm 1}=1/2$, ${\hat{f}}_2=-i/4$, and ${\hat{f}}_{-2}=i/4$. This explains why the real and complex lists look different while representing the same function.

A trigonometric polynomial is a finite linear combination of complex exponentials $e^{inx}$, or equivalently of sines and cosines. If it vanishes on an interval, all its coefficients vanish.

The Leibniz formula is the conditionally convergent alternating series