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For conjugate exponents $p,q$,with the analogous inequality ${\sum }_n|x_n{y}_n|\le \Vert x{\Vert }_p\Vert y{\Vert }_q$ for sequences.

The continuous dual space $X^{\ast }$ of a topological vector space is the vector space of its continuous linear functionals. Unlike the algebraic dual, it requires that a small change in an input produce a small change in its scalar measurement. For a normed vector space, this is equivalent to a finite boundfor all vectors $x$ and some constant $C\ge 0$. The least such constant is the operator norm of $\phi$.

This is the continuity-restricted subspace of the algebraic dual space; in infinite dimension the two can differ.

For the linear functional $\phi (x,y)=2x-y$ on ${\mathbb{R}}^2$ with its Euclidean norm, the Cauchy-Schwarz inequality givesEquality holds on the unit vector $(2,-1)/\sqrt{5}$, so its operator norm is exactly $\sqrt{5}$. The same estimate controls measurement error: $|\phi (v+h)-\phi (v)|\le \sqrt{5}\Vert h{\Vert }_2$. The linear functional diagram shows which changes are entirely invisible to this measurement; its kernel of a linear map consists of the changes with zero response. All linear functionals on a finite-dimensional real or complex normed vector space are continuous, but discontinuous functional detected by vanishing sequence tails shows why the restriction matters in infinite dimension.

Translations of a compactly supported bump can form an infinite family separated by a fixed sup-norm distance, disproving compactness.

C0(X) consists of continuous functions whose values become arbitrarily small outside compact sets.

A Sobolev space measures a function together with specified weak derivatives in Lp spaces. For an open set $U\subseteq {\mathbb{R}}^n$, an integer $k\ge 0$ and $1\le p\le \infty$, defineThe derivatives are understood through integration by parts against test functions, rather than by requiring classical differentiability at every point. The Sobolev norm makes this a Banach space; for $p=2$, write $H^k(U)=W^{k,2}(U)$, a Hilbert space.^[ref]

For example, $u(x)=|x|$ on $(-1,1)$ belongs to $H^1$ even though it is not differentiable at zero: its weak derivative is $\mathrm{sgn}(x)$ and both functions are square-integrable. A step with a jump at zero does not belong to $H^1(-1,1)$, because its distributional derivative is a Dirac delta, which is not an L2 function. This distinction admits corners while excluding jumps at first-order square-integrable regularity.

The Fourier transform extends the square-integrable scale to every real $s$. With a unitary Fourier convention, $H^s({\mathbb{R}}^n)$ consists of tempered distributions satisfyingFor integer $s\ge 0$, this norm is equivalent to the weak-derivative Sobolev norm. Such spaces are natural for weak formulations of partial differential equations: a finite gradient energy can exist even when the prospective solution lacks classical second derivatives.

The theorem supplies solvability for a coercive weak formulation.

A bounded bilinear form $a$ on a real Hilbert space $H$ that is a coercive bilinear form represents every bounded linear functional uniquely: for each $\ell \in H^{\prime }$, there is a unique $u\in H$ with $a(u,v)=\ell (v)$ for every $v\in H$.

For a proof, the Riesz representation theorem gives a bounded linear operator $T$ and a vector $g$ with $a(u,v)=⟨Tu,v⟩$ and $\ell (v)=⟨g,v⟩$. Coercivity and the Cauchy-Schwarz inequality give $\Vert Tu\Vert \ge \alpha \Vert u\Vert$, so $T$ is injective and its image is closed: convergence of $T{u}_n$ makes $u_n$ a Cauchy sequence, whose limit supplies the image limit. If $v$ belongs to the orthogonal complement of the image, then $a(u,v)=0$ for every $u$; setting $u=v$ and using coercivity forces $v=0$. Thus the image is also dense and equals $H$. There is consequently a unique $u$ with $Tu=g$, and $\Vert u\Vert \le \Vert \ell \Vert /\alpha$. The complex version uses a bounded sesquilinear form with $\mathrm{Re}a(u,u)\ge \alpha \Vert u{\Vert }^2$.

The closed unit ball of a dual Banach space is compact in the weak-star topology.

Every bounded linear functional on a linear subspace of a real normed space extends to the whole space without increasing its norm.

The numerical range is $W(A)=\{⟨Ax,x⟩:x\in D(A),\Vert x\Vert =1\}$.

A bounded linear operator $T:X\to Y$ between normed spaces is a compact operator if the image of the unit ball has compact closure in $Y$. Equivalently, every bounded input sequence has a subsequence whose images converge in the target norm.

On ${\ell }^2$, $(Tx{)}_n=x_n/n$ is compact. Retaining its first $N$ coordinates gives a finite-rank operator, and the operator-norm error is $1/(N+1)$. By contrast, the identity on infinite-dimensional ${\ell }^2$ is not compact: it maps the standard unit vectors to themselves, and no subsequence is norm convergent. Compactness is therefore a stronger property than boundedness, reflecting a form of finite-dimensional approximability in this example.

A functional $F:X\to (-\infty ,\infty ]$ on a normed vector space has weak lower semicontinuity when each sublevel set $\{x:F(x)\le a\}$ is closed in the weak topology. Equivalently, for every weakly convergent net $x_i\rightharpoonup x$,The analogous condition for sequences defines sequential weak lower semicontinuity. In a general topological space, sequence tests alone need not characterize a closed set.

For a convex functional on a normed vector space, lower semicontinuity in norm topology is equivalent to weak lower semicontinuity: convex sublevel sets are norm closed exactly when they are weakly closed, by the Hahn-Banach separation theorem. In this convex setting the sequential condition also suffices, because norm-convergent sequences converge weakly and the norm topology is metrizable. In particular, the norm has weak lower semicontinuity. Its negative is not: in ${\ell }^2$, $e_n\rightharpoonup 0$, while $-\Vert e_n\Vert =-1<0=-\Vert 0\Vert$. The correct semicontinuity is what preserves an objective bound when a minimizing sequence has only a weak limit.

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 $X\times Y$.

For $1\le p\le \infty$, a diagonal linear operator on the l-p sequence space is $A(x_n)=(a_n{x}_n)$. It is bounded exactly when ${\sup }_n|a_n|<\infty$, and then $\Vert A\Vert ={\sup }_n|a_n|$: the upper bound follows coordinatewise, and the coordinate unit vectors give the reverse inequality.

The linear operator is injective exactly when every $a_n$ is nonzero. In that case its inverse on its range is bounded exactly when ${\inf }_n|a_n|>0$. If a diagonal coefficient vanishes, there is no inverse on the full domain; one can instead restrict the domain to sequences supported on the nonzero coordinates. Thus a statement about an inverse must specify injectivity or this restriction.

The linear operator is compact exactly when $a_n\to 0$. Truncating to the first $N$ coordinates gives a finite-rank operator with error ${\sup }_{n>N}|a_n|$. Conversely, if infinitely many $|a_n|\ge \epsilon$, the corresponding images $a_n{e}_n$ stay separated in norm and have no convergent subsequence. The example $a_n=1/n$ is a compact injective linear operator with an unbounded inverse, a basic model of an ill-posed inverse problem.

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.