GATE Mathematics Functional Analysis Flashcards

24 sample cards from the Functional Analysis deck

Sampled from the end of the deck, so these are different cards from the ones shown on the syllabus page.

1. When is a Hilbert space separable, in terms of its orthonormal basis?

   A Hilbert space is separable iff it has a countable orthonormal basis. All separable infinite-dimensional Hilbert spaces are isometrically isomorphic to $\ell^{2}$.

2. State the Projection Theorem for a closed subspace of a Hilbert space.

   If $M$ is a closed subspace of a Hilbert space $H$, then $H=M\oplus M^{\perp}$: every $x\in H$ has a unique decomposition $x=m+m^{\perp}$ with $m\in M$, $m^{\perp}\in M^{\perp}$. The map $P:x\mapsto m$ is the orthogonal projection onto $M$.

3. State the closest-point (best approximation) property characterizing the orthogonal projection $m=Px$.

   For $x\in H$ and closed convex $M$, there is a unique $m\in M$ minimizing $\|x-m\|$. When $M$ is a closed subspace, $m=Px$ is characterized by $x-m\perp M$, i.e. $\langle x-m,y\rangle=0$ for all $y\in M$.

4. List the defining algebraic properties of an orthogonal projection $P$ onto a closed subspace.

   $P$ is linear, idempotent ($P^{2}=P$), self-adjoint ($P^{*}=P$), with $\|P\|=1$ (if $P\neq 0$), range $M$ closed, kernel $M^{\perp}$, and $I-P$ is the orthogonal projection onto $M^{\perp}$.

5. For a subset $S$ of a Hilbert space, what is $S^{\perp}$ and what is $(S^{\perp})^{\perp}$?

   $S^{\perp}=\{x\in H:\langle x,s\rangle=0\ \forall s\in S\}$ is always a closed subspace. $(S^{\perp})^{\perp}=\overline{\operatorname{span}}\,S$. In particular if $M$ is a closed subspace, $M^{\perp\perp}=M$.

6. State the Riesz Representation Theorem for bounded linear functionals on a Hilbert space.

   For every $f\in H^{*}$ there exists a unique $y\in H$ such that $f(x)=\langle x,y\rangle$ for all $x\in H$, and $\|f\|=\|y\|$. The map $f\mapsto y$ is a conjugate-linear isometric bijection of $H^{*}$ onto $H$.

7. What does the Riesz Representation Theorem imply about a Hilbert space and its dual (reflexivity, self-duality)?

   $H$ is (conjugate-linearly) isometrically isomorphic to its dual $H^{*}$, so $H$ is self-dual and every Hilbert space is reflexive.

8. State the Lax-Milgram theorem (Riesz-type representation for bounded coercive sesquilinear forms).

   If $a:H\times H\to\mathbb{C}$ is a bounded ($|a(x,y)|\leq C\|x\|\|y\|$) and coercive ($\operatorname{Re}a(x,x)\geq c\|x\|^{2}$, $c>0$) sesquilinear form, then for each $f\in H^{*}$ there is a unique $u\in H$ with $a(u,v)=f(v)$ for all $v\in H$.

9. Define the adjoint $T^{*}$ of a bounded operator $T$ on a Hilbert space and give its key norm identity.

   $T^{*}$ is the unique bounded operator satisfying $\langle Tx,y\rangle=\langle x,T^{*}y\rangle$ for all $x,y$. Its existence follows from Riesz representation. Key identities: $\|T^{*}\|=\|T\|$ and $\|T^{*}T\|=\|T\|^{2}$ (the C*-identity).

10. Define a self-adjoint operator and state the key fact about its inner-product values.

    A bounded operator $T$ is self-adjoint if $T=T^{*}$, i.e. $\langle Tx,y\rangle=\langle x,Ty\rangle$ for all $x,y$. Then $\langle Tx,x\rangle\in\mathbb{R}$ for all $x$, and $\|T\|=\sup_{\|x\|=1}|\langle Tx,x\rangle|$.

11. Define a compact operator between Banach spaces.

    A linear operator $T:X\to Y$ is compact if the image $T(B_{X})$ of the closed unit ball has compact closure in $Y$; equivalently, every bounded sequence $\{x_{n}\}$ has a subsequence with $\{Tx_{n}\}$ convergent.

12. State the Spectral Theorem for a compact self-adjoint operator on a Hilbert space.

    If $T$ is a compact self-adjoint operator on a Hilbert space $H$, then $H$ has an orthonormal basis $\{e_{n}\}$ consisting of eigenvectors of $T$, with real eigenvalues $\lambda_{n}$, and $Tx=\sum_{n}\lambda_{n}\langle x,e_{n}\rangle e_{n}$. The nonzero eigenvalues form a finite set or a sequence $\lambda_{n}\to 0$, each of finite multiplicity.

13. For a compact self-adjoint operator, what are the properties of its eigenvalues and eigenvectors?

    Eigenvalues are real; eigenvectors for distinct eigenvalues are orthogonal; each nonzero eigenvalue has finite-dimensional eigenspace; the nonzero eigenvalues form at most a sequence converging to $0$; $\|T\|=\max_{n}|\lambda_{n}|$ equals the largest $|\lambda|$.

14. State the result that guarantees a nonzero compact self-adjoint operator has an eigenvalue.

    For a nonzero compact self-adjoint $T$, at least one of $\pm\|T\|$ is an eigenvalue; equivalently there is an eigenvalue $\lambda$ with $|\lambda|=\|T\|$. This is the starting point of the spectral theorem's proof.

15. Define the spectrum $\sigma(T)$ and the resolvent set of a bounded operator on a complex Banach space.

    $\sigma(T)=\{\lambda\in\mathbb{C}:T-\lambda I\text{ is not invertible in }B(X)\}$. The resolvent set is its complement $\rho(T)$, on which $(T-\lambda I)^{-1}$ exists and is bounded. For $X\neq\{0\}$ complex, $\sigma(T)$ is nonempty, compact, and contained in $\{|\lambda|\leq\|T\|\}$.

16. Classify the spectrum into point, continuous, and residual parts.

    Point spectrum: $\lambda$ with $T-\lambda I$ not injective (eigenvalues). Continuous spectrum: $T-\lambda I$ injective with dense, non-closed range. Residual spectrum: $T-\lambda I$ injective with range not dense.

17. Describe the spectrum of a compact operator on an infinite-dimensional Banach space (Riesz-Schauder theory).

    The spectrum is a countable set with $0$ as its only possible limit point; every nonzero spectral value is an eigenvalue of finite multiplicity, and $0\in\sigma(T)$ when $\dim X=\infty$.

18. Compare the spectrum of a self-adjoint operator with that of a unitary operator on a Hilbert space.

    For a bounded self-adjoint operator $\sigma(T)\subseteq\mathbb{R}$ (and residual spectrum is empty). For a unitary operator $U$ ($U^{*}U=UU^{*}=I$), $\sigma(U)\subseteq\{z\in\mathbb{C}:|z|=1\}$, the unit circle.

19. State the spectral radius and Gelfand's spectral radius formula.

    The spectral radius is $r(T)=\max\{|\lambda|:\lambda\in\sigma(T)\}$, and $r(T)=\lim_{n\to\infty}\|T^{n}\|^{1/n}\leq\|T\|$. For a self-adjoint (or normal) operator, $r(T)=\|T\|$.

20. Compare strong and weak convergence with norm convergence in a Hilbert space.

    Norm (strong) convergence $\|x_{n}-x\|\to 0$ implies weak convergence $\langle x_{n},y\rangle\to\langle x,y\rangle$ for all $y$, but not conversely. With weak convergence plus $\|x_{n}\|\to\|x\|$, one recovers norm convergence (Radon-Riesz property). An orthonormal sequence $e_{n}\rightharpoonup 0$ weakly but not in norm.

21. State the relationship $\ker(T^{*})=(\operatorname{ran} T)^{\perp}$ and the resulting orthogonal decomposition.

    For bounded $T$ on a Hilbert space, $\ker(T^{*})=(\operatorname{ran} T)^{\perp}$ and $\overline{\operatorname{ran} T}=(\ker T^{*})^{\perp}$, giving $H=\overline{\operatorname{ran} T}\oplus\ker T^{*}$.

22. State the Fredholm Alternative for $T=I-K$ with $K$ compact.

    For compact $K$ and $\lambda\neq 0$, either $(\lambda I-K)x=y$ has a unique solution for every $y$ (so $\lambda I-K$ is bijective), or the homogeneous equation $(\lambda I-K)x=0$ has a nontrivial solution; in the latter case $\dim\ker(\lambda I-K)=\dim\ker(\lambda I-K^{*})<\infty$.

23. Compare the Hahn-Banach Theorem with the Open Mapping, Closed Graph, and Uniform Boundedness theorems in terms of completeness requirements.

    Hahn-Banach requires no completeness — it holds in any normed (even vector) space. The Open Mapping, Closed Graph, and Uniform Boundedness theorems are 'Baire category' theorems requiring completeness (Banach spaces) of the relevant space(s).

24. Define a normal operator and relate the classes normal, self-adjoint, and unitary.

    $T$ is normal if $TT^{*}=T^{*}T$. Self-adjoint ($T=T^{*}$) and unitary ($T^{*}=T^{-1}$) operators are both special cases of normal operators. A bounded operator is normal iff $\|Tx\|=\|T^{*}x\|$ for all $x$.

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