Not to scare you, but list of requirements for a first course in functional analysis is rather long:

• Basic theorems of metric spaces including, but not limited to:
• The Baire category theorem
• $\ell^p$ is complete
• Arzelà-Ascoli (how else will you show that an operator is compact?)
• Measure theory --- or at least be ready to accept that you have to learn some while reading functional analysis. Because the Riesz representation theorem essentially says that for a big class of "reasonable" spaces, continuous linear functionals and measures are the same. In other words a lot of the theory will make no sense without at least knowing some measure theory.
• Topology. If you want to go beyond Banach spaces and study Fréchet spaces. The continuous dual of a Fréchet space that is not a Banach space is not necessarily metrisable --- and you get to work with multiple different topologies on your spaces (weak, strong, weak-*)

If that doesn't scare you off, I can recommend the information-dense "Introduction to Functional Analysis" by Reinhold Meise and Dietmar Vogt. ISBN 0-19-851485-9.

And when I say dense i mean very dense. It clocks in at a modest 437 pages, yet in a late undergraduate course in functional analysis we covered less than a third of that book (plus some notes on convexity) in a semester.

As for Rudin's Real & Complex Analysis: it's a great book, but I don't know if I'd really call it a book on functional analysis. I'd say it's on analysis in general --- hence the title.

UPDATE: If you find that you need to brush up on real analysis, Terence Tao has notes for 3 courses on his webpage: Real Analysis 245A (in progress at the time of writing), 245B and 245C. Actually I think I can highly recommend the entirety of his webpage.

Not to scare you, but list of requirements for a first course in functional analysis is rather long:

• Basic theorems of metric spaces including, but not limited to:
• The Baire category theorem
• $\ell^p$ is complete
• Arzelà-Ascoli (how else will you show that an operator is compact?)
• Measure theory --- or at least be ready to accept that you have to learn some while reading functional analysis. Because the Riesz representation theorem essentially says that for a big class of "reasonable" spaces, continuous linear functionals and measures are the same. In other words a lot of the theory will make no sense without at least knowing some measure theory.
• Topology. If you want to go beyond Banach spaces and study Fréchet spaces. The continuous dual of a Fréchet space that is not a Banach space is not necessarily metrisable --- and you get to work with multiple different topologies on your spaces (weak, strong, weak-*)

If that doesn't scare you off, I can recommend the information-dense "Introduction to Functional Analysis" by Reinhold Meise and Dietmar Vogt. ISBN 0-19-851485-9.

And when I say dense i mean very dense. It clocks in at a modest 437 pages, yet in a late undergraduate course in functional analysis we covered less than a third of that book (plus some notes on convexity) in a semester.

As for Rudin's Real & Complex Analysis: it's a great book, but I don't know if I'd really call it a book on functional analysis. I'd say it's on analysis in general --- hence the title.

UPDATE: If you find that you need to brush up on real analysis, Terence Tao has notes for 3 courses on his webpage: Real Analysis 245A (in progress at the time of writing), 245B and 245C. Actually I think I can highly recommend the entirety of his webpage.

Not to scare you, but list of requirements for a first course in functional analysis is rather long:

• Basic theorems of metric spaces including, but limited to:
• The Baire category theorem
• $\ell^p$ is complete
• Arzelà-Ascoli (how else will you show that an operator is compact?)
• Measure theory --- or at least be ready to accept that you have to learn some while reading functional analysis. Because the Riesz representation theorem essentially says that for a big class of "reasonable" spaces, continuous linear functionals and measures are the same. In other words a lot of the theory will make no sense without at least knowing some measure theory.
• Topology. If you want to go beyond Banach spaces and study Fréchet spaces. The continuous dual of a Fréchet space that is not a Banach space is not necessarily metrisable --- and you get to work with multiple different topologies on your spaces (weak, strong, weak-*)

If that doesn't scare you off, I can recommend the information-dense "Introduction to Functional Analysis" by Reinhold Meise and Dietmar Vogt. ISBN 0-19-851485-9.

And when I say dense i mean very dense. It clocks in at a modest 437 pages, yet in a late undergraduate course in functional analysis we covered less than a third of that book (plus some notes on convexity) in a semester.

As for Rudin's Real & Complex Analysis: it's a great book, but I don't know if I'd really call it a book on functional analysis. I'd say it's on analysis in general --- hence the title.

Not to scare you, but list of requirements for a first course in functional analysis is rather long:

• Basic theorems of metric spaces including, but not limited to:
• The Baire category theorem
• $\ell^p$ is complete
• Arzelà-Ascoli (how else will you show that an operator is compact?)
• Measure theory --- or at least be ready to accept that you have to learn some while reading functional analysis. Because the Riesz representation theorem essentially says that for a big class of "reasonable" spaces, continuous linear functionals and measures are the same. In other words a lot of the theory will make no sense without at least knowing some measure theory.
• Topology. If you want to go beyond Banach spaces and study Fréchet spaces. The continuous dual of a Fréchet space that is not a Banach space is not necessarily metrisable --- and you get to work with multiple different topologies on your spaces (weak, strong, weak-*)

If that doesn't scare you off, I can recommend the information-dense "Introduction to Functional Analysis" by Reinhold Meise and Dietmar Vogt. ISBN 0-19-851485-9.

And when I say dense i mean very dense. It clocks in at a modest 437 pages, yet in a late undergraduate course in functional analysis we covered less than a third of that book (plus some notes on convexity) in a semester.

As for Rudin's Real & Complex Analysis: it's a great book, but I don't know if I'd really call it a book on functional analysis. I'd say it's on analysis in general --- hence the title.

UPDATE: If you find that you need to brush up on real analysis, Terence Tao has notes for 3 courses on his webpage: Real Analysis 245A (in progress at the time of writing), 245B and 245C. Actually I think I can highly recommend the entirety of his webpage.