GATE Mathematics Functional Analysis Syllabus

Functional Analysis syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Functional Analysis in GATE Mathematics, not a summary of it.

1. 1

   Normed Linear Spaces

   4 topics

   • Hahn-Banach Theorem
   • Open Mapping Theorem
   • Closed Graph Theorem
   • Principle of Uniform Boundedness
2. 2

   Inner-Product Spaces

   1 topic

   • Hilbert Spaces
     • Orthonormal Bases
     • Projection Theorem
     • Riesz Representation Theorem
     • Spectral Theorem for Compact Self-Adjoint Operators

Functional Analysis flashcards for GATE Mathematics

23 of 50 cards from the Functional Analysis deck — real questions with worked answers.

1. State the Hahn-Banach Theorem (dominated extension form) for a real vector space.

   Let $X$ be a real vector space and $p:X\to\mathbb{R}$ a sublinear functional (i.e. $p(x+y)\leq p(x)+p(y)$ and $p(\alpha x)=\alpha p(x)$ for $\alpha\geq 0$). If $f$ is a linear functional on a subspace $M\subseteq X$ with $f(x)\leq p(x)$ for all $x\in M$, then $f$ extends to a linear functional $F$ on $X$ with $F(x)\leq p(x)$ for all $x\in X$.

2. State the norm-preserving (normed space) version of the Hahn-Banach Theorem.

   If $X$ is a normed space, $M\subseteq X$ a subspace, and $f\in M^{*}$, then there exists $F\in X^{*}$ with $F|_{M}=f$ and $\|F\|=\|f\|$.

3. Hahn-Banach corollary: for a nonzero vector $x_{0}$ in a normed space $X$, what functional is guaranteed to exist?

   There exists $F\in X^{*}$ with $\|F\|=1$ and $F(x_{0})=\|x_{0}\|$. Consequently $\|x_{0}\|=\sup_{\|F\|\leq 1}|F(x_{0})|$, so $X^{*}$ separates points of $X$.

4. What is the complex (Bohnenblust-Sobczyk) form of the Hahn-Banach Theorem's domination condition?

   For a complex space with a seminorm $p$, a linear functional $f$ on $M$ with $|f(x)|\leq p(x)$ extends to $F$ on $X$ with $|F(x)|\leq p(x)$ for all $x\in X$.

5. Hahn-Banach separation: state the geometric (separating hyperplane) consequence for disjoint convex sets.

   If $A,B$ are disjoint nonempty convex subsets of a normed space, $A$ open, then there exist $f\in X^{*}$ (real) and $\gamma\in\mathbb{R}$ with $\operatorname{Re}f(a)<\gamma\leq\operatorname{Re}f(b)$ for all $a\in A,\,b\in B$. If $A$ compact and $B$ closed (both convex, disjoint) they can be strictly separated.

6. State the Open Mapping Theorem.

   If $X,Y$ are Banach spaces and $T:X\to Y$ is a bounded linear surjection, then $T$ is an open map (the image of every open set is open).

7. What is the Bounded Inverse Theorem and how does it follow from the Open Mapping Theorem?

   If $T:X\to Y$ is a bounded linear bijection between Banach spaces, then $T^{-1}$ is bounded. It follows because an open bijection has a continuous inverse.

8. What completeness hypotheses are essential for the Open Mapping Theorem, and what category-style ingredient drives its proof?

   Both $X$ and $Y$ must be Banach (complete). The proof uses the Baire Category Theorem applied to $Y=\bigcup_{n}\overline{T(nB_{X})}$.

9. State the Closed Graph Theorem.

   If $X,Y$ are Banach spaces and $T:X\to Y$ is a linear map whose graph $\{(x,Tx):x\in X\}$ is closed in $X\times Y$, then $T$ is bounded (continuous).

10. Express the closed graph condition sequentially.

    The graph of $T$ is closed iff: whenever $x_{n}\to x$ in $X$ and $Tx_{n}\to y$ in $Y$, it follows that $y=Tx$. For a continuous map one only knows $x_{n}\to x\Rightarrow Tx_{n}\to Tx$; the closed graph theorem supplies the converse direction needed for boundedness.

11. How are the Open Mapping Theorem and the Closed Graph Theorem related?

    They are equivalent (each can be derived from the other). The Closed Graph Theorem is typically proved by applying the Open Mapping/Bounded Inverse Theorem to the projection $\pi_{X}:G(T)\to X$ from the (closed, hence Banach) graph $G(T)$.

12. State the Principle of Uniform Boundedness (Banach-Steinhaus Theorem).

    Let $X$ be a Banach space, $Y$ a normed space, and $\{T_{\alpha}\}\subseteq B(X,Y)$ a family of bounded operators. If $\sup_{\alpha}\|T_{\alpha}x\|<\infty$ for every $x\in X$ (pointwise boundedness), then $\sup_{\alpha}\|T_{\alpha}\|<\infty$ (uniform boundedness).

13. Banach-Steinhaus consequence: if $T_{n}\to T$ pointwise (strongly) with each $T_{n}\in B(X,Y)$ and $X$ Banach, what can you conclude about $T$?

    The limit $T$ is linear and bounded, with $\|T\|\leq\liminf_{n}\|T_{n}\|$, because $\sup_{n}\|T_{n}\|<\infty$ by the uniform boundedness principle.

14. Which space must be complete in the Uniform Boundedness Principle, and what tool proves it?

    The domain $X$ must be a Banach space (the codomain need only be normed). The proof rests on the Baire Category Theorem.

15. Define an inner product space and the norm it induces.

    An inner product space is a vector space $H$ with a map $\langle\cdot,\cdot\rangle:H\times H\to\mathbb{C}$ that is linear in one argument, conjugate-symmetric ($\langle x,y\rangle=\overline{\langle y,x\rangle}$), and positive definite ($\langle x,x\rangle\geq 0$, $=0$ iff $x=0$). It induces the norm $\|x\|=\sqrt{\langle x,x\rangle}$.

16. Define a Hilbert space.

    A Hilbert space is an inner product space that is complete with respect to the norm $\|x\|=\sqrt{\langle x,x\rangle}$ induced by its inner product.

17. State the Cauchy-Schwarz inequality and the condition for equality.

    $|\langle x,y\rangle|\leq\|x\|\,\|y\|$, with equality iff $x$ and $y$ are linearly dependent.

18. State the parallelogram law and what it characterizes among normed spaces.

    $\|x+y\|^{2}+\|x-y\|^{2}=2\|x\|^{2}+2\|y\|^{2}$. A norm comes from an inner product iff it satisfies the parallelogram law (Jordan-von Neumann theorem).

19. State the polarization identity for a complex inner product space.

    $\langle x,y\rangle=\frac{1}{4}\sum_{k=0}^{3} i^{k}\,\|x+i^{k}y\|^{2}=\frac{1}{4}\big(\|x+y\|^{2}-\|x-y\|^{2}+i\|x+iy\|^{2}-i\|x-iy\|^{2}\big)$.

20. State the Pythagorean theorem in an inner product space for orthogonal vectors.

    If $x\perp y$ (i.e. $\langle x,y\rangle=0$), then $\|x+y\|^{2}=\|x\|^{2}+\|y\|^{2}$. More generally for pairwise orthogonal $x_{1},\dots,x_{n}$: $\|\sum x_{k}\|^{2}=\sum\|x_{k}\|^{2}$.

21. Define an orthonormal set and an orthonormal basis of a Hilbert space.

    A set $\{e_{\alpha}\}$ is orthonormal if $\langle e_{\alpha},e_{\beta}\rangle=\delta_{\alpha\beta}$. It is an orthonormal basis (complete orthonormal set) if its closed linear span is all of $H$, equivalently if the only vector orthogonal to every $e_{\alpha}$ is $0$.

22. State Bessel's inequality for an orthonormal set $\{e_{n}\}$.

    For any $x\in H$, $\sum_{n}|\langle x,e_{n}\rangle|^{2}\leq\|x\|^{2}$. The numbers $\langle x,e_{n}\rangle$ are the Fourier coefficients of $x$.

23. State Parseval's identity and what it characterizes.

    For an orthonormal set $\{e_{\alpha}\}$, $\|x\|^{2}=\sum_{\alpha}|\langle x,e_{\alpha}\rangle|^{2}$ for all $x\in H$ iff $\{e_{\alpha}\}$ is an orthonormal basis. Equivalently $x=\sum_{\alpha}\langle x,e_{\alpha}\rangle e_{\alpha}$.

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