Summary of Key Theorems

Statement. Let $X$ be a normed vector space and $Y \subseteq X$ a subspace. If $f : Y \to \mathbb{R}$ is a bounded linear functional, then there exists an extension $F : X \to \mathbb{R}$ such that $F|_Y = f$ and $\|F\| = \|f\|$.

Intuition. The theorem guarantees that we can extend linear functionals from a subspace to the whole space without increasing the norm. This is fundamental for duality theory.

Corollaries. For any nonzero $x \in X$ there exists $f \in X^*$ with $\|f\| = 1$ and $f(x) = \|x\|$. The natural embedding $X \to X^{**}$ is isometric.

Statement. If $T : X \to Y$ is a bounded surjective linear operator between Banach spaces, then $T$ is an open map (the image of every open set is open).

Intuition. Surjectivity plus completeness forces the operator to be well-behaved topologically. A key consequence is that a bijective bounded linear operator has a bounded inverse.

Corollary (Inverse Mapping Theorem). If $T : X \to Y$ is a bijective bounded linear operator between Banach spaces, then $T^{-1}$ is bounded.

Statement. Let $T : X \to Y$ be a linear operator between Banach spaces. Then $T$ is bounded if and only if its graph $\Gamma(T) = \{(x, Tx) : x \in \mathrm{dom}(T)\}$ is closed in $X \times Y$.

Intuition. This theorem provides a powerful way to prove boundedness: instead of checking continuity directly, one checks that whenever $x_n \to x$ and $Tx_n \to y$, then $y = Tx$.

Common Pitfall. The domain and codomain must both be Banach spaces. A linear operator $T : C[0,1] \to C[0,1]$ defined by $Tf = f''$ has a closed graph in $C[0,1] \times C[0,1]$, but its domain $C^2[0,1]$ is not complete under the sup norm. The theorem does not apply.

Statement (Banach-Steinhaus). Let $\{T_\alpha\}_{\alpha \in A}$ be a family of bounded linear operators from a Banach space $X$ to a normed space $Y$. If $\sup_\alpha \|T_\alpha x\| < \infty$ for each $x \in X$, then $\sup_\alpha \|T_\alpha\| < \infty$.

Intuition. Pointwise boundedness implies uniform boundedness. This is a consequence of the Baire category theorem and is often used to prove that certain operators are unbounded.

Application. If a sequence of bounded operators converges pointwise, the limit operator is bounded and the sequence is uniformly bounded in norm.

Statement. Let $H$ be a Hilbert space. For every bounded linear functional $f : H \to \mathbb{R}$, there exists a unique vector $y \in H$ such that $f(x) = \langle x, y \rangle$ for all $x \in H$, and $\|f\| = \|y\|$.

Intuition. In a Hilbert space, the dual space is isometrically isomorphic to the space itself. Every functional is just an inner product with some vector.

Consequence. The map $y \mapsto \langle \cdot, y \rangle$ is an isometric isomorphism $H \cong H^*$, so Hilbert spaces are self-dual.

Statement. If $T : H \to H$ is a compact self-adjoint operator on a Hilbert space $H$, then there exists an orthonormal basis of $H$ consisting of eigenvectors of $T$. The corresponding eigenvalues are real and converge to zero.

Intuition. Compact self-adjoint operators are diagonalisable: they behave like infinite-dimensional symmetric matrices. This generalises the finite-dimensional spectral theorem.

Application. Sturm-Liouville problems, integral equations with symmetric kernels, and the quantum mechanical position and momentum operators all involve spectral decompositions.

Statement. Let $T : X \to X$ be a compact operator on a Banach space $X$. For any nonzero $\lambda \in \mathbb{C}$, either $\lambda I - T$ is invertible with bounded inverse, or $\lambda$ is an eigenvalue of $T$ with finite multiplicity.

Intuition. For compact perturbations of the identity, the only possible obstructions to invertibility are eigenvalues. The null space and range have complementary dimensions.

Application. This theorem is essential for solving integral equations of the second kind: $f - \lambda K f = g$, where $K$ is a compact integral operator.

Statement. The closed unit ball of the dual space $X^*$ of a normed space $X$ is compact in the weak* topology.

Intuition. Even though the unit ball in an infinite-dimensional space is never compact in the norm topology, it becomes compact in the weak* topology. This provides a powerful existence principle for optimisation problems.

Application. Existence of minimisers in the calculus of variations: every bounded sequence in the dual has a weak* convergent subsequence.

These six core theorems are deeply interrelated. The Hahn-Banach, Open Mapping, Closed Graph, and Uniform Boundedness principles are often called the four pillars of functional analysis. All rely on the Baire category theorem and completeness.

The Riesz Representation and Spectral theorems are specific to Hilbert spaces, where the geometry of inner products provides stronger structure. The Fredholm Alternative bridges compact operator theory with spectral theory. Banach-Alaoglu uses weak* topology to recover compactness lost in infinite dimensions.

1. Let $X = C[0,1]$ and $f(x) = x(0)$. Use the Hahn-Banach theorem to extend $f$ to a functional on $L^\infty[0,1]$ with the same norm.
2. Suppose $T : X \to Y$ is a linear operator between Banach spaces with closed graph. Show that $T$ is bounded. Why is the converse trivial?
3. Give an example of a family of operators $\{T_n\}$ such that $\|T_n x\|$ is bounded for each $x$ but $\|T_n\| \to \infty$.
4. Verify that the operator $(Tf)(x) = \int_0^1 \sin(xy) f(y)\,dy$ on $L^2[0,1]$ is compact and self-adjoint. What does the spectral theorem tell you about its eigenfunctions?

The six core theorems of functional analysis give the subject its power: extension of functionals (Hahn-Banach), stability of surjectivity (Open Mapping), automatic continuity from graph closure (Closed Graph), uniform bounds from pointwise bounds (Uniform Boundedness), concrete dual representation (Riesz), and diagonalisation of compact operators (Spectral).