Spaces §

From a number to a size §

In finite dimensions, a vector $x = (x_1, \dots, x_n)$ has a natural size, the Euclidean length $\sqrt{x_1^2 + \cdots + x_n^2}$. A norm is exactly the generalization of “length” to any vector space: it assigns a non-negative size, respects scaling, and obeys the triangle inequality. The triangle inequality is what makes the norm a genuine notion of distance: $d(x, y) = \|x - y\|$ is a metric.

Completeness is the real content §

The interesting property is not the norm but completeness. A sequence is Cauchy if its terms get arbitrarily close to each other: $\|x_n - x_m\| \to 0$ as $n, m \to \infty$. A space is complete if every such sequence actually converges to a point inside the space.

Example. The space $c_{00}$ of finitely supported sequences with the $\ell^2$ norm is not complete: the sequence $x^{(n)} = (1, 1/2, \dots, 1/n, 0, 0, \dots)$ is Cauchy but converges to $(1, 1/2, 1/3, \dots)$, which has infinitely many nonzero terms and so is not in $c_{00}$. Its completion is the Banach space $\ell^2$.

The workhorse examples §

• $C([a,b])$ with the sup norm $\|f\|_\infty$ is a Banach space.
• $L^p$ for $1 \leq p \leq \infty$ (equivalence classes of functions with finite $p$-norm) is a Banach space; its construction rests on the measure-theoretic integral, see Measure theory.
• $\ell^p$ (sequences with finite $p$-norm) is a Banach space.

The inner product adds geometry §

A norm gives you lengths and distances; an inner product adds angles and orthogonality. The key identity is the parallelogram law,

$\|x + y\|^2 + \|x - y\|^2 = 2\|x\|^2 + 2\|y\|^2,$

which holds exactly for norms coming from an inner product. It is the reason a Hilbert space “looks like” Euclidean space: you can project, decompose into orthogonal pieces, and talk about angles.

Example. $L^2([a,b])$ has the inner product $\langle f, g \rangle = \int_a^b f \overline{g}$. The space $\ell^2$ has $\langle x, y \rangle = \sum x_n \overline{y_n}$.

Orthogonal projection and the Pythagorean theorem §

Given a closed subspace $M \subset H$, every $x \in H$ splits uniquely as $x = m + h$ with $m \in M$ and $h \perp M$. This is the infinite-dimensional version of dropping a perpendicular. It yields the Pythagorean theorem $\|x\|^2 = \|m\|^2 + \|h\|^2$, and it is the engine behind Fourier series: the partial sums are orthogonal projections onto finite-dimensional subspaces.

Bases and the Riesz representation theorem §

In a Hilbert space, every continuous linear functional $\varphi$ has the form $\varphi(x) = \langle x, y \rangle$ for a unique $y \in H$. This is the Riesz representation theorem, and it is what makes Hilbert spaces self-dual: the space and its dual are canonically the same. It fails in general Banach spaces, which is the dividing line between the two settings.

Why look at the dual §

The dual collects all the “measuring devices” on a space — every continuous way to extract a single number from a vector. Two spaces that are hard to distinguish directly often have very different duals, so the dual is a powerful invariant.

Example. The dual of $\ell^1$ is $\ell^\infty$: a continuous linear functional on $\ell^1$ is given by pairing with a bounded sequence. The dual of $L^p$ for $1 \leq p < \infty$ is $L^q$ where $1/p + 1/q = 1$ (Hölder’s inequality).

Hahn–Banach: extending functionals §

The Hahn–Banach theorem says a continuous linear functional defined on a subspace can be extended to the whole space without increasing its norm. Concretely, it guarantees that the dual is “large enough”: for every nonzero $x \in X$ there is a functional $\varphi$ with $\varphi(x) = \|x\|$ and $\|\varphi\| = 1$. This is the reason the dual norm recovers the original norm: $\|x\| = \sup_{\|\varphi\| \leq 1} |\varphi(x)|$.

What reflexivity means §

Every Banach space embeds into its double dual via $J$, but $J$ need not be surjective. Reflexivity says it is: no “extra” functionals appear when you dualize twice. Reflexive spaces are the well-behaved ones for existence arguments — they are the setting where weak compactness (below) gives you convergent subsequences.

Examples. $\ell^p$ and $L^p$ for $1 < p < \infty$ are reflexive. $\ell^1$, $\ell^\infty$, and $C([0,1])$ are not reflexive. Every finite-dimensional space is reflexive.

The weak topology and why reflexivity pays off §

The weak topology on $X$ is the coarsest topology making every $\varphi \in X^*$ continuous; a sequence $x_n \rightharpoonup x$ (weak convergence) means $\varphi(x_n) \to \varphi(x)$ for every $\varphi$. The key fact: in a reflexive space, every bounded sequence has a weakly convergent subsequence (the Eberlein–Šmulian / Banach–Alaoglu machinery). This is the workhorse behind existence theorems for PDEs and optimization — see Applications.

Problem. Let $H$ be a Hilbert space and $M \subset H$ a closed subspace. Prove that $H = M \oplus M^\perp$ (direct sum), and explain why this decomposition is the reason Hilbert spaces are reflexive.