Are there any good introductory textbook to cover all these topics?

[zli034]

Linear Spaces
Norms and inner products
Holder’s inequality
Minkowski’s inequality
Normed linear spaces
Cauchy sequences and complete spaces
Banach spaces
Reitz representation theorem
Hilbert spaces
Orthogonal bases
Generalized Fourier expansions

Lebesgue Measure and Integration
Sigma fields
Lebesgue outer measure
Lebesgue measurability of sets
Borel sets
Measurable functions
Lebesgue’s Theorem
Egoroff’s Theorem
Lebesgue Integration
Bounded convergence theorem
Fatou’s lemma
Monotone convergence theorem
Dominated convergence theorem
Absolute continuity

 

[micromass]

Real analysis by Yeh covers most of these topics very well!