Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 4 / 22I (Linear Analysis)

The Riesz representation theorem says that for every bounded linear functional $f$ on a Hilbert space $H$, there is a unique $z\in H$ such thatand $\Vert f\Vert =\Vert z\Vert$.

For $T\in L(H)$ and fixed $y\in H$, the mapis a bounded linear functional, withRiesz therefore gives a unique vector, denoted $T^{\ast }y$, such thatfor every $x$. Uniqueness in the representation theorem shows that $y\mapsto T^{\ast }y$ is linear. Moreover,so the adjoint operator belongs to $L(H)$ and $\Vert T^{\ast }\Vert \le \Vert T\Vert$.

If $(e_j)$ is an orthonormal basis and $A=(a_{ij})$ is the matrix of $T$, then the matrix of $T^{\ast }$ isits entries are $\overline{a_{ji}}$.

The spectral theorem for compact Hermitian operators says that the nonzero spectrum of a compact Hermitian $T$ consists of real eigenvalues of finite multiplicity, with zero as the only possible accumulation point. Eigenvectors for distinct eigenvalues are orthogonal, and an orthonormal basis of ${\ell }^2$ can be chosen from eigenvectors together with a basis of $\ker T$. Thuswhere the nonzero eigenvalues are repeated according to multiplicity and ${\lambda }_n\to 0$ if there are infinitely many.

DefineEach $T_N$ is finite-rank and Hermitian, andHence every compact Hermitian operator is a norm limit of finite-rank Hermitian operators, by finite-rank truncation of a compact Hermitian operator.

Let $P_N$ be the orthogonal projection of ${\ell }^2$ onto the span of its first $N$ standard basis vectors. Then $P_{N}y\to y$ for every $y\in {\ell }^2$, and $\Vert I-P_N\Vert \le 1$.

Let $T$ be compact and letwhere $B$ is the closed unit ball. The set $K$ is compact. Given $\epsilon >0$, choose a finite $\epsilon /3$-net $y_1,\dots ,y_m$ in $K$. Pointwise convergence lets us choose $N$ such thatfor every $j$. If $y\in K$ and $\Vert y-y_j\Vert <\epsilon /3$, thenThus $P_N\to I$ uniformly on $K$, andEach $P_{N}T$ has image in an $N$-dimensional space, so it has finite rank. This coordinate-projection approximation of a compact operator proves the result.