Is the book Real Analysis by H.L. Royden, 4th edition, suitable for two introductory functional analysis courses comprising the following topics?
First Course:
Metric spaces: A quick review, convergence and completeness, completion
Normed Spaces: Linear spaces, normed spaces, relation between normed and metric spaces, Banach spaces, bounded and continuous linear operators and functionals, dual spaces, finite-dimensional normed spaces, F. Riesz lemma, the Hahn-Banach (HB) theorems (for real and complex vector spaces and for normed spaces), the open-mapping theorem, the closed-graph theorem, the uniform boundedness principle and its applications
The Banach fixed-point theorem with applications to algebraic, differential, and integral equations
Second Course:
Hilbert Spaces: Inner product space & Hilbert space, orthogonal & orthonormal sets, orthogonal complements, Gram-Schmidt orthogonalization process, the Riesz representation theorem for representation of functionals on a Hilbert spaces in terms of the inner product, weak and weak* convergence
Finite-Dimensional Spectral Theory: The definition of spectrum of an operator and some examples, spectral properties of self-adjoint operators, the spectral-mapping theorem for finite-dimensional Hilbert spaces
Can Royden be used to learn the above topics or teach those topics to a group of students who are not very sharp and whose mathematical maturity isn't very good?
How about Royden being used together with the book Introductory Functional Analysis With Applications by Erwine Kreyszig?
The issue with Kreyszig is that
(1) it doesn't take measure theory into consideration at all,
(2) it is not rigorous enough in some of its proofs,
and
(3) it is unnecessarily wordy at some other places
Maybe a second edition of Kreyszig will take care of these issues.
But is it a good idea to use Royden as a textbook for both the Functional Analysis (I & II) courses (consisting of the above-mentioned topics) as well as for the Measure Theory (I & II) courses learnt by or taught to the same audience?
Last but not least, links please to web pages of courses taught using either or both of the above two books, especially the former!