Prerequisites & Notation

Before You Begin

This chapter builds the infinite-dimensional framework needed for RF imaging. It assumes the following background. If any item feels unfamiliar, revisit the linked material before proceeding.

Linear algebra: vector spaces, inner products, eigenvalues, SVD (Telecom Book, Chapter 1) (Review ch01)
Self-check: Can you compute the SVD of a $3 \times 2$ complex matrix and interpret the singular values geometrically?
Calculus: sequences, series, limits, continuity in $\mathbb{R}$ ; uniform vs. pointwise convergence
Self-check: Can you prove that a Cauchy sequence in $\mathbb{R}$ converges, and give an example of a sequence that is Cauchy in $\mathbb{Q}$ but not in $\mathbb{Q}$ ?
Measure theory and Lebesgue integration at the level of "almost everywhere" and the dominated convergence theorem
Self-check: Can you state the dominated convergence theorem and give an example where pointwise convergence does not imply $L^1$ convergence?
Complex numbers: modulus, conjugate, polar form, $e^{j\theta}$ ; basic complex analysis at the undergraduate level
Self-check: Can you evaluate $\int_0^1 e^{j 2\pi n x}\,dx$ for integer $n$ ?
Basic set theory and proof techniques: induction, contradiction, epsilon-delta arguments
Self-check: Can you write an $\varepsilon$ - $\delta$ proof of continuity for a given function?

Notation for This Chapter

Symbols introduced in this chapter. All imaging-specific symbols use $\backslash$ ntn{} tokens where a notation key exists for the book. See also the NGlobal Notation Table master table.

Symbol	Meaning	Introduced
$\mathcal{V}, \mathcal{W}, \mathcal{H}$	Generic normed, Banach, or Hilbert spaces	s01
$\\|\cdot\\|$	Norm on a vector space (context determines the space)	s01
$\\|\cdot\\|_{L^p}$	Lebesgue $L^p$ norm on function spaces	s01
$L^p(\Omega)$	Lebesgue space of $p$ -integrable functions on $\Omega$	s01
$\ell^p$	Sequence space with $p$ -summable entries	s01
$\langle f, g \rangle$	Inner product (linear in first argument)	s02
$L^2(\Omega)$	Hilbert space of square-integrable functions	s02
$\{e_n\}_{n=1}^\infty$	Orthonormal basis (ONB)	s02
$\hat{f}$	Orthogonal projection of $f$ onto a subspace	s02
$\mathcal{A}: \mathcal{V} \to \mathcal{W}$	Bounded linear operator	s03
$\\|\mathcal{A}\\|_{\mathrm{op}}$	Operator norm: $\sup_{\\|f\\|=1}\\|\mathcal{A}f\\|$	s03
$\mathcal{A}^*$	Adjoint operator	s03
$\sigma_n(\mathcal{A})$	Singular values of operator $\mathcal{A}$ , $\sigma_1 \geq \sigma_2 \geq \cdots \to 0$	s04
$\{(\sigma_n, v_n, u_n)\}$	Singular system: right singular functions, left singular functions	s04
$\mathcal{D}'(\Omega)$	Space of distributions (continuous linear functionals on test functions)	s05
$H^s(\Omega)$	Sobolev space of order $s$ (weak derivatives in $L^2$ )	s05
$G(\mathbf{r}, \mathbf{r}^\prime)$	Green's function of the Helmholtz operator	s05
$\mathcal{A}: \mathcal{X} \to \mathcal{Y}$	Forward operator (maps scene to measurements)	s06
$\Omega$	Target / scene region in 3-D space	s06
$c(\mathbf{r})$	Complex reflectivity (scene) function at position $\mathbf{r}$	s06
$\mathbf{A}$	Discretized sensing matrix	s06
$\mathbf{w}$	Noise vector in the observation model	s06

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