1.

**Bounded inverse theorem**(a bijective bounded linear operator T from one Banach space to another has bounded inverse )

2.

**Hahn-Banach theorem**(to extend bounded linear functions on a subspace/v.s. to the whole space)

3.

**Hellinger-Toeplitz theorem**(...symmetric operator on a Hilbert space is bounded)

4.

**Riesz representation theorem**(to map Hilbert space H towards its dual space H*)

Fréchet-Riesz theorem: Φ:

*H*→

*H**defined by Φ(

*x*) = φ

_{x}is an isometric (anti-) isomorphism. Quantum Mechanics: every ket has a corresponding bra ,

5.

**Banach-Steinhaus theorem**(Uniform boundedness principle)

For continuous linear operators whose domain is a Banach space, pointwise boundedness is

equivalent to uniform boundedness in operator norm.

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Top 5 Theorems in Real Analysis and Measure Theory

1.

**Bolzano–Weierstrass theorem**(each bounded sequence in R^n has a convergent subsequence)

2.

**Fatou–Lebesgue theorem**(a chain of inequalities relating Lebesgue integrals)

Dominated convergence theorem (a special case of Fatou–Lebesgue theorem.)

3. Fundamental theorem of calculus, Mean value theorem, Taylor's theorem (for Engineers like me):

4.

**Heine–Borel theorem**(A subset of a metric space is compact iff it is complete and totally bounded)

5.

**Uniform limit theorem**(the uniform limit of any sequence of continuous functions is continuous.)

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Top 5 Inequalities in Real Analysis (with minimum words:)

1.

**Hölder's inequality**:

(A special case) Cauchy–Schwarz inequality:

^{}

2.

**Minkowski inequality**:

(A special case) Triangle inequality:

3.

**Chebyshev's inequality**:

4.

**Inequality of arithmetic and geometric means**:

5.

**Jensen's inequality**:

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