Functional analysis Study Guide

📖 Core Concepts Functional analysis – study of vector spaces equipped with limits (norms, inner products, topologies) and the linear maps that respect those structures. Normed vector space – a vector space $V$ with a function $\|\,\cdot\,\|:V\to[0,\infty)$ satisfying the norm axioms. Banach space – a normed space that is complete (every Cauchy sequence converges). Inner‑product space – a vector space with a bilinear (or sesquilinear) form $\langle\cdot,\cdot\rangle$ that induces a norm $\|x\|=\sqrt{\langle x,x\rangle}$. Hilbert space – a complete inner‑product space; every Hilbert space is a Banach space, but not every Banach space is Hilbert. Fréchet space – a locally convex topological vector space that is metrizable and complete; it may lack a norm. Dual space $X^{}$ – the set of all continuous linear functionals $f:X\to\mathbb{K}$ (scalar field). Bidual $X^{}$ – the dual of the dual; every Banach space embeds isometrically into its bidual via the canonical map. --- 📌 Must Remember Banach ↔ complete norm; Hilbert ↔ complete inner product. $L^{p}(X,\mu)$ ($1\le p\le\infty$) and $\ell^{p}$ are canonical Banach spaces. $\ell^{2}$ is the (unique up to isomorphism) separable infinite‑dimensional Hilbert space. Hahn–Banach: any bounded linear functional on a subspace extends to the whole space without increasing its norm. Open Mapping Theorem: a surjective continuous linear operator $T:X\to Y$ between Banach spaces sends open sets to open sets. Closed Graph Theorem: $T$ is continuous iff $\{(x,Tx):x\in X\}$ is closed in $X\times Y$. Uniform Boundedness Principle: pointwise bounded family $\{T\alpha\}$ of continuous linear operators on a Banach space is uniformly bounded in operator norm. Spectral theorem (bounded self‑adjoint): $T = U^{}MfU$ with $U:H\to L^{2}(\Omega,\mu)$ unitary and $Mf$ multiplication by a real bounded measurable $f$. Spectral theorem (bounded normal): same representation, but $f$ may be complex‑valued. --- 🔄 Key Processes Applying Hahn–Banach Identify a sublinear function $p$ on $V$. Verify $f(u)\le p(u)$ for all $u$ in the subspace $U$. Extend $f$ to $F$ on $V$ with $F(x)\le p(x)$ and $\|F\|=\|f\|$. Checking Continuity via Closed Graph Write the graph $GT=\{(x,Tx)\}\subset X\times Y$. Prove $GT$ is closed (use limit arguments). Conclude $T$ is continuous. Using Uniform Boundedness Show $\sup{\alpha}\|T\alpha x\|<\infty$ for each fixed $x\in X$. Conclude $\sup{\alpha}\|T\alpha\|<\infty$. Spectral Decomposition of a Bounded Self‑Adjoint $T$ Find a unitary $U:H\to L^{2}(\Omega,\mu)$. Determine $f\in L^{\infty}(\Omega,\mu;\mathbb{R})$ such that $T = U^{}MfU$. --- 🔍 Key Comparisons Banach vs. Hilbert Banach: any complete normed space; may lack inner product, orthonormal bases. Hilbert: norm comes from an inner product; always has orthonormal bases. Finite‑dimensional vs. Infinite‑dimensional Hilbert Finite: reduces to linear algebra; every basis is orthonormal after Gram‑Schmidt. Infinite separable: all isomorphic to $\ell^{2}$; bases are countable. Self‑adjoint vs. Normal (bounded) Self‑adjoint: $T=T^{}$, spectral measure is real‑valued. Normal: $TT^{}=T^{}T$, spectral measure may be complex‑valued. Normed vs. Fréchet spaces Normed: single norm gives topology. Fréchet: defined by a countable family of seminorms; may have no single norm. --- ⚠️ Common Misunderstandings “Every Banach space has an orthogonal basis.” – False; orthogonal (or orthonormal) bases exist only in Hilbert spaces. “Hahn–Banach gives a unique extension.” – Extensions are generally not unique; only existence is guaranteed. “Open Mapping works for any linear map.” – Requires surjectivity and both domain and codomain to be Banach. “All Hilbert spaces are separable.” – Only those with a countable orthonormal basis are; non‑separable Hilbert spaces exist (larger cardinalities). “The spectral theorem applies to unbounded operators.” – The version stated applies only to bounded self‑adjoint or normal operators. --- 🧠 Mental Models / Intuition Banach = “complete” version of $\mathbb{R}^{n}$ – think of Cauchy sequences as “never‑ending” paths that must land somewhere. Hilbert = infinite‑dimensional Euclidean space – you can still talk about angles and orthogonal projections. Dual space = “measurement devices.” Every functional picks out a single number (like a probe) from a vector. Canonical embedding $X\hookrightarrow X^{}$ – imagine every vector being identified with the way all measurements act on it. --- 🚩 Exceptions & Edge Cases Bidual non‑surjectivity – The canonical embedding $X\to X^{}$ may fail to be onto (non‑reflexive spaces). Lack of orthogonal bases in Banach spaces – No analogue of Gram‑Schmidt; classification is harder. Spectral theorem limitation – Only for bounded self‑adjoint or normal operators; unbounded operators need a different (spectral) framework. Non‑separable Hilbert spaces – Have orthonormal bases of higher cardinalities; not isomorphic to $\ell^{2}$. --- 📍 When to Use Which | Situation | Tool / Theorem | Decision Rule | |-----------|----------------|---------------| | Need to extend a bounded functional from a subspace | Hahn–Banach | Sublinear bound $p$ available → extend without norm increase. | | Proving a linear operator is continuous but explicit norm estimate is hard | Closed Graph | Show graph is closed → continuity follows. | | You have a surjective bounded operator and want to know image of open sets | Open Mapping | Operator is onto → images of opens are open. | | Dealing with a family $\{T\alpha\}$ pointwise bounded on $X$ | Uniform Boundedness | Pointwise bound $\forall x$ → uniform bound on $\|T\alpha\|$. | | Operator is bounded and self‑adjoint (or normal) on a Hilbert space | Spectral Theorem | Represent as multiplication operator after a unitary change of basis. | | Working in a complete normed space and need a “nice” basis for expansions | Hilbert space (if inner product present) | Use orthonormal basis; otherwise settle for Schauder bases (not covered). | --- 👀 Patterns to Recognize “Bounded linear functional on a subspace” → think Hahn–Banach. “Surjective linear map between Banach spaces” → Open Mapping likely applies. “Graph of $T$ closed in $X\times Y$” → continuity via Closed Graph. “Family $\{T\alpha\}$ pointwise bounded” → Uniform Boundedness yields a common norm bound. “Self‑adjoint (or normal) bounded operator” → look for a spectral representation $U^{}MfU$. --- 🗂️ Exam Traps Distractor: “Every Banach space has an orthonormal basis.” – Confuses Banach with Hilbert. Distractor: “Hahn–Banach guarantees a unique norm‑preserving extension.” – Only existence, not uniqueness. Distractor: “If $T$ is linear and its graph is closed, $T$ must be bounded and invertible.” – Closed graph gives boundedness, not invertibility. Distractor: “Open Mapping Theorem works for non‑surjective operators.” – Surjectivity is essential. Distractor: “Spectral theorem applies to any bounded operator.” – Requires self‑adjoint or normal. Distractor: “A Banach space is always reflexive (i.e., $X = X^{}$).” – False; many Banach spaces are non‑reflexive. ---