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A bounded operator is compact when it maps the unit ball to a relatively compact set.

A functional $F$ is weakly lower semicontinuous when $x_j\rightharpoonup x$ implies $F(x)\le {\mathrm{liminf}}_{j}F(x_j)$. Convex integral functionals with suitable growth have this property, which lets weak limits of minimizing sequences remain minimizers.

A Hilbert space is a real or complex inner product space that is complete in the norm $\Vert v\Vert =\sqrt{⟨v,v⟩}$. Completeness means that every Cauchy sequence has a limit in the space: if its terms become arbitrarily close in this norm, they converge to an allowed vector. The inner product supplies lengths and orthogonality; completeness ensures that limiting approximation procedures do not leave the space.

Finite-dimensional Euclidean spaces are examples, but the same geometry also works for sequences and functions. In the l2 sequence space,In the L2 space on $[-1,1]$ with Lebesgue measure,These formulas use the convention linear in the first argument. Elements of the L2 space are equivalence classes of square-integrable functions agreeing almost everywhere, so zero distance really identifies the same element. A vector can therefore be a whole function, and its squared length can measure an integrated squared error.

For example, approximating $f(x)=x^2$ by $a+bx$ is a nearest-point problem in this Hilbert space. The allowed approximations form the linear span of $1,x$, and the orthogonal projection is the constant $1/3$. Its residual $x^2-1/3$ is orthogonal to both allowed directions because its integral and its inner product with $x$ vanish. The geometry therefore identifies the best approximation without searching through every pair of coefficients.

This is useful whenever a complicated vector must be represented by a tractable collection of directions. Fourier series approximate functions with orthogonal oscillations; the Parseval identity relates the sum of squared coefficients to the squared norm. Least-squares solutions approximate attainable data using a linear operator. The Hilbert projection theorem explains why a nearest element exists and is unique for a nonempty closed convex set. Distinct application settings share the same geometry of useful components and orthogonal residuals. Casey Rodriguez's MIT lectures develop these connections between Hilbert spaces, Fourier series, and nearest-point problems.^[ref]

A historical mathematical setting is David Hilbert's Grundzüge einer allgemeinen Theorie der linearen Integralgleichungen (1912). Its preface states that it reprints six communications published during 1904–1910 on linear integral equations. This provides a concrete source for the integral-equation setting associated with the name, while the abstract definition above incorporates later developments.^[ref]

A linear subspace is dense exactly when the only continuous functional vanishing on it is zero, by Hahn-Banach separation.

If $A$ is invertible and $\Vert A^{-1}E\Vert <1$, then $A+E$ is invertible by the Neumann series. Similar iterative corrections prove stability of surjectivity.

A bounded injective map between Banach spaces is bounded below exactly when its image is closed, by the bounded inverse theorem applied to its image.

The graph of $T:X\to Y$ is the linear subspace $\Gamma (T)=\{(x,Tx):x\in X\}$ of $X\times Y$.

A diagonal operator on ${\ell }^p$ acts coordinatewise as $(x_n)\mapsto (a_n{x}_n)$. It is bounded when $(a_n)$ is bounded, while its inverse on its image is bounded only when the nonzero $a_n$ are bounded away from zero.

A bounded surjective linear map between Banach spaces is open. Baire category first puts a ball in the closure of an image; iterative correction removes the closure.

A point-separating real subalgebra of $C(X)$ that contains the constants is uniformly dense when $X$ is compact Hausdorff. Polynomial approximation of the square root makes its closure a lattice; finite maxima and minima then turn pointwise interpolation into uniform approximation.

A normed vector space is a vector space equipped with a norm, whose induced metric is $d(x,y)=\Vert x-y\Vert$. The norm must be positive on nonzero vectors, scale by the absolute value of a scalar, and satisfy the triangle inequality. These conditions make distance compatible with addition and scaling.

On $C([0,1])$, both the supremum norm $\Vert f{\Vert }_{\infty }={\max }_{0\le x\le 1}|f(x)|$ and the L2 norm $\Vert f{\Vert }_2=({\int }_0^1|f(x){|}^{2}dx{)}^{1/2}$ are genuine norms. A nonzero continuous function stays nonzero on an interval of positive length, so its squared integral cannot vanish. For $f(x)=x$, they give $1$ and $1/\sqrt{3}$, respectively. Always $\Vert f{\Vert }_2\le \Vert f{\Vert }_{\infty }$, but the shrinking peaks below show that no uniform reverse bound exists. Unlike the finite-dimensional equivalence of norms, infinite-dimensional vector spaces can support inequivalent norm topologies.

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 with a complete submultiplicative norm, $\Vert ab\Vert \le \Vert a\Vert \Vert b\Vert$.

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.