GATE Mathematics Real Analysis Syllabus

Real Analysis syllabus — full chapter and topic list

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

1. 1

   Metric spaces

   overview

   Examined as a single unit within Real Analysis — no further topic split in the official outline.

2. 2

   Connectedness

   overview

   Examined as a single unit within Real Analysis — no further topic split in the official outline.

3. 3

   Compactness

   overview

   Examined as a single unit within Real Analysis — no further topic split in the official outline.

4. 4

   Completeness

   overview

   Examined as a single unit within Real Analysis — no further topic split in the official outline.

5. 5

   Sequences and series of functions

   2 topics

   • Uniform convergence
   • Ascoli-Arzela theorem
6. 6

   Weierstrass approximation theorem

   overview

   Examined as a single unit within Real Analysis — no further topic split in the official outline.

7. 7

   Contraction mapping principle

   overview

   Examined as a single unit within Real Analysis — no further topic split in the official outline.

8. 8

   Power series

   overview

   Examined as a single unit within Real Analysis — no further topic split in the official outline.

9. 9

   Differentiation of functions of several variables

   overview

   Examined as a single unit within Real Analysis — no further topic split in the official outline.

10. 10

    Inverse and Implicit function theorems

    overview

    Examined as a single unit within Real Analysis — no further topic split in the official outline.

11. 11

    Lebesgue measure on the real line

    overview

    Examined as a single unit within Real Analysis — no further topic split in the official outline.

12. 12

    Measurable functions

    overview

    Examined as a single unit within Real Analysis — no further topic split in the official outline.

13. 13

    Lebesgue integral

    3 topics

    • Fatou’s lemma
    • Monotone convergence theorem
    • Dominated convergence theorem

Real Analysis flashcards for GATE Mathematics

20 of 50 cards from the Real Analysis deck — real questions with worked answers.

1. Define pointwise convergence of a sequence of functions $f_n \to f$ on a set $E$.

   $f_n \to f$ pointwise on $E$ if for every $x \in E$ and every $\varepsilon > 0$, there exists $N = N(\varepsilon, x)$ such that $|f_n(x) - f(x)| < \varepsilon$ for all $n \geq N$. The index $N$ may depend on $x$.

2. Define uniform convergence of $f_n \to f$ on a set $E$.

   $f_n \to f$ uniformly on $E$ if for every $\varepsilon > 0$ there exists $N = N(\varepsilon)$ (independent of $x$) such that $|f_n(x) - f(x)| < \varepsilon$ for all $n \geq N$ and all $x \in E$.

3. State the sup-norm criterion for uniform convergence.

   $f_n \to f$ uniformly on $E$ if and only if $$\lim_{n \to \infty} \sup_{x \in E} |f_n(x) - f(x)| = 0.$$

4. State the Cauchy criterion for uniform convergence.

   $f_n$ converges uniformly on $E$ if and only if for every $\varepsilon > 0$ there exists $N$ such that $\sup_{x \in E}|f_n(x) - f_m(x)| < \varepsilon$ for all $m, n \geq N$.

5. If $f_n \to f$ uniformly and each $f_n$ is continuous on $E$, what can you conclude about $f$?

   The limit $f$ is continuous on $E$. Uniform convergence preserves continuity: $$\lim_{x \to x_0}\lim_{n\to\infty} f_n(x) = \lim_{n\to\infty}\lim_{x \to x_0} f_n(x).$$

6. Give the standard example showing pointwise but not uniform convergence on $[0,1]$.

   $f_n(x) = x^n$ on $[0,1]$ converges pointwise to $f(x)=0$ for $x\in[0,1)$ and $f(1)=1$. Convergence is not uniform since $\sup_{[0,1]}|f_n(x)-f(x)| = 1$ for all $n$, and the limit is discontinuous.

7. State the theorem on integrating a uniformly convergent sequence on $[a,b]$.

   If $f_n \to f$ uniformly on $[a,b]$ and each $f_n$ is Riemann integrable, then $f$ is integrable and $$\lim_{n\to\infty}\int_a^b f_n\,dx = \int_a^b f\,dx = \int_a^b \lim_{n\to\infty} f_n\,dx.$$

8. State the theorem on differentiating a sequence of functions term by term.

   If $f_n$ are differentiable on $[a,b]$, $f_n(x_0)$ converges for some $x_0$, and $f_n'$ converges uniformly to $g$, then $f_n$ converges uniformly to some $f$ with $f' = g$, i.e. $\left(\lim f_n\right)' = \lim f_n'$.

9. State the Weierstrass M-test for uniform convergence of a series $\sum f_n$.

   If $|f_n(x)| \leq M_n$ for all $x \in E$ and $\sum_{n=1}^{\infty} M_n < \infty$, then $\sum_{n=1}^{\infty} f_n$ converges uniformly (and absolutely) on $E$.

10. State Dini's theorem on uniform convergence.

    If $K$ is compact, $f_n \to f$ pointwise on $K$, each $f_n$ and $f$ are continuous, and $f_n$ is monotone in $n$ (e.g. $f_n(x) \downarrow f(x)$) for each $x$, then $f_n \to f$ uniformly on $K$.

11. Why does uniform convergence imply pointwise convergence but not conversely?

    Uniform convergence requires a single $N(\varepsilon)$ working for all $x$, which in particular works for each fixed $x$, giving pointwise convergence. The converse fails because pointwise $N$ may depend on $x$ with no uniform bound (e.g. $x^n$ on $[0,1)$).

12. Show with an example that pointwise convergence does not allow interchange of limit and integral.

    Let $f_n(x) = n^2 x(1-x)^n$ on $[0,1]$. Then $f_n \to 0$ pointwise, but $\int_0^1 f_n\,dx \to 1 \neq 0$. The convergence is not uniform, so limit and integral cannot be swapped.

13. Define equicontinuity of a family $\mathcal{F}$ of functions at a point $x_0$.

    $\mathcal{F}$ is equicontinuous at $x_0$ if for every $\varepsilon > 0$ there exists $\delta > 0$ such that $|f(x) - f(x_0)| < \varepsilon$ for all $f \in \mathcal{F}$ whenever $|x - x_0| < \delta$. The same $\delta$ works for every member of the family.

14. Define uniform equicontinuity of a family $\mathcal{F}$ on a set $E$.

    $\mathcal{F}$ is uniformly equicontinuous on $E$ if for every $\varepsilon > 0$ there exists $\delta > 0$ such that for all $f \in \mathcal{F}$ and all $x, y \in E$ with $|x-y| < \delta$, we have $|f(x) - f(y)| < \varepsilon$. Here $\delta$ is independent of both $f$ and the points.

15. Define uniform boundedness of a family $\mathcal{F}$ of functions on $E$.

    $\mathcal{F}$ is uniformly bounded on $E$ if there exists $M > 0$ such that $|f(x)| \leq M$ for all $f \in \mathcal{F}$ and all $x \in E$. A single bound $M$ works for the entire family.

16. State the Ascoli–Arzelà theorem.

    Let $K$ be a compact metric space and $\mathcal{F} \subseteq C(K)$. Then $\mathcal{F}$ is relatively compact (every sequence has a uniformly convergent subsequence) if and only if $\mathcal{F}$ is uniformly bounded and equicontinuous.

17. What are the two hypotheses required by the Ascoli–Arzelà theorem to extract a uniformly convergent subsequence?

    (1) Uniform boundedness of the family, and (2) equicontinuity of the family. Together, on a compact domain, they guarantee a uniformly convergent subsequence.

18. In Ascoli–Arzelà, what is the conclusion about a sequence $\{f_n\}$ that is uniformly bounded and equicontinuous on compact $K$?

    There exists a subsequence $\{f_{n_k}\}$ that converges uniformly on $K$ (to a continuous limit). This is sequential compactness in $(C(K), \|\cdot\|_\infty)$.

19. How does Ascoli–Arzelà characterize compact subsets of $C(K)$ for compact $K$?

    A subset $\mathcal{F} \subseteq C(K)$ is compact if and only if it is closed, uniformly bounded, and equicontinuous. (Relative compactness drops the closedness requirement.)

20. Why does a Lipschitz bound with a common constant give equicontinuity?

    If $|f(x) - f(y)| \leq L|x-y|$ for all $f$ in the family with the same $L$, then choosing $\delta = \varepsilon / L$ gives $|f(x)-f(y)| < \varepsilon$ for all members. A uniform bound on derivatives, $|f'| \leq L$, suffices.

See more Real Analysis flashcards →

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