A “really” intro level text on functional analysis, at the master/advanced undergrad level.

It is very well written – clear & smooth, and a good portion of the exercises are elementary so that one can reinforce her understanding of definitions and theorems without getting stuck too often.

Highly recommended for anyone trying to learn functional analysis.

## CH1 Preliminaries

### A bunch of inequalities

Let be conjugate exponents, i.e., . Here are complex numbers.

**Young’s inequality**:
.

**Holder’s inequality**:

(can let
if the resulting series are convergent; also, this follows
from Young)

**Holder’s inequality, integral version**:

( are Lebesgue measurable functions defined on
and the integrals are all finite by assumption)

**Cauchy Schwarz inequality**: setting
in Holder’s inequality.

**Minkowski’s inequality**: if
, then

(can let if the resulting series are convergent; proof mostly relies on triangle inequality. Note that if , sign of Minkowski reversed)

Minkowski also has integral version:

if , then we have .

### Zorn’s lemma

**poset**:
,
is a partial order (consistent ranking, but may be
incomplete)

**chain**:
,
is a linear order (complete: any two elements can be
ranked)

**least element of
**: an element
s.t.
(so
must be able to be compared with all elements of
)

**lower bound of
**: an element
s.t.
( must be able to be compared with all elements of
)

**minimal element of
**: an element
s.t.
( doesn’t have to be able to be compared with every element
of
, just make sure that if it can for some
)

**Zorn’s lemma**: If
is a poset s.t. every chain in
has an upper bound, then
contains a maximal element.

As an application, can show every abstract vector space has a Hamel basis.

for a vector space over a field , A subset of vectors in is a

**Hamel basis**for if is linearly independent and spansproof is actually short. But seems Schauder basis introduced later is the focus.

## CH2 Metric Spaces

**separable metric space**

having a countable dense subset.

some standard topology concepts in baby Rudin Ch2. Then some general topology definitions and results. I remind myself those that I’m starting to forget.

**neighborhood**: given metric space , a neighborhood of is a subset of s.t. it contains an open ball , for some .

**Cantor intersection property**: if
is complete, then for a sequence of contracting nonempty
closed subsets of
, there exists
s.t.

First, perhaps the book should mention that the sequence of sets has the property that their diameters tend to 0:

Then, according to Wiki, this is actually a variant of “Cantor’s intersection theorem”, which is stated in general topology (and another version which is stated for subsets of .

This can be used to show that is uncountable. If it is, we write it as a sequence . Then we find a interval with length , such that it doesn’t contain . Then, divide this interval into 3 closed subintervals of equal length. Then at least one of these subintervals doesn’t contain . Keep doing this, we construct a sequence of closed intervals, contracting in diameter, but it contains an element in the limit which is not a term of the sequence .

**Continuous mapping between metric spaces**

1. For a mapping
between metric spaces, continuity at a point defined in
the
way is equivalent to continuity defined via sequences.

- Continuity of the mapping on the whole domain is equivalent to the notion that inverse image of any open set is open.

**isometry (mapping between metric spaces)**:

a mapping between metric spaces that preserves distance:
is isometry if
. (by def, this is a continuous mapping, and injective)

**isometric metric spaces**

is isometric to
if there exists a **bijective isometry** from
onto
. (note bijectivity implies existence of inverse isometric
map, so if can just say
are isometric to each other)

**extension of isometry to the whole space**

For two complete metric spaces, if they each have a dense subset, and
there is an isometry (mapping) between them, then there exists an
isometry between the two metric spaces themselves, and the isometric
mappings are consistent with the one associated with dense subsets.

thus, perhaps we can say that if two metric spaces are isometric, then their Banach completions are isometric.

**completing metric spaces** first, completion of
means
such that:

1.
is complete 2. $(X,d) is isometric to a dense subset of

for example, is a complete of .

Can show that any metric space has a completion (long proof), and it is somewhat unique in the sense that if we have two completions of the same metric space, then the two completions are isometric to each other.

**Baire Category Theorem, v1**

Let
be a sequence of **dense** open sets in a
complete metric space
. Then,
.

**Baire Category Theorem, v2**

Let
be a complete metric space. If
, where
are closed sets, then at least one of these sets contain
an open ball.

The reason this is called “category thm” is that we call a subset of a topological space “nowhere dense” if its closure contains no interior points. (the interior of of a topological space is defined as the union of all subsets of that are open in )

A set is 1st category if it is a countable union of nowhere dense sets. If not, the set is called 2nd category. Then, Baire v2 says that every complete metric space is of 2nd category.

This can actually be used to “prove” is uncountable by writing , and using the fact that is a closed set.

Similarly, this can be used to show that is not a complete metric space.

### Compactness

define a metric space to be “compact” if every sequence in it has a convergent subsequence.

(later, a topological space is defined “compact” if every open cover has finite subcover.)

A compact set in a metric space is closed and bounded, but the
converse is only true for
. For general metric spaces, The Borel-Lebesgue theorem
says that it is compact **iff** every open cover
of
contains a finite subcover.

**continuous mapping & compact domain** (metric
space results)

1. If a mapping is continuous, then take any compact set, its image
under this mapping is compact.

- If a mapping is continuous, and is compact, then is bounded, and attains its maximum value.

later, analogous results for topological spaces.

### Topological Spaces

**a more general structure than that of a metric
space**.

This book uses the notation . Given this topology, let . Then is a topology on , and is called the *subspace topology**. Always remember that a topology is just a collection of sets we call “open”.

**Hausdorff topological space**: if every two distinct
points in the space have disjoint neighborhoods.

**base/basis of topology** a family of open subsets of
s.t. every open subset of
is the union of sets belonging to this family. Say this
family generates the topology. Many topologies are more easily defined
in terms of a generating set.

**characterizing topological base** Given a topology
, can show that
is its base iff:

1.

2.
,
, there exists a set
s.t.

Intuitively, first make sure the base can cover all of ; then, make sure it doesn’t generate slices (intersections) too thin.

**As an example, in any metric space, the collection of open
balls is a base of the metric topology on
**.

**Continuous mapping between topological spaces**

If are topological spaces, a mapping is continuous at if for any neighborhood of the point , there is a neighborhood of s.t. .

**A mapping between topological spaces are continuous iff the inverse image under of every open set in is an open set in**.is discrete topology iff every mapping from into any topological space is continuous.

**Homeomorphic spaces** two topological spaces
are homeomorphic if there is a bijection
s.t.
and
are continuous.

**Metrizable topological space**

if the topology is homeomorphic to a metric space.

note that if two metric spaces are isometric, then they are homeomorphic as topological spaces. But not the converse. For example, consider and , with the usual Euclidean metric. It is easy to construct a linear monotone bijective mapping from one to another, hence homeomorphic; yet, you can never get in the space , so the two spaces can not be isometric.

**compactness, topological spaces**
is defined to be compact if every open cover of
contains a finite sub-cover.
is compact if
is compact in the subspace topology.

**2 results for continuous mapping & compact
domain** (topological space results) Let
be a continuous mapping. Then for any compact
,
is a compact set in
.

If is compact, then a continuous function is bounded and attains its maximum.

**product topology**

Given
and
, a topology on
is defined by taking as a base the collection of sets of
the form
, where
for
.

(note that you actually need to show the collection of sets so
constructed indeed is a base)

## CH3 Special Spaces (Topological/Metric/Normed vector spaces)

Topological, metric, and normed vector spaces will be introduced. together with operators on them, they are the main focus of Functional Analysis.

This chapter introduces some important special cases, and establish their topological and metric properties.

First we should clarify the difference between metric space and vector space.

metric space in general have no notion of addition and scalar multiplication (note we are not looking at the special case of normed space, which induces a metric, but also is by definition a vector space)

vector space in general have no notion of distance.

**topological vector space** defined as a vector space
over field
, endowed with a topology such that vector addition and
scalar multiplication are continuous functions. That is:

for any vector , for any neighborhood of this vector, there exists neighborhood of and neighborhood of s.t.

for any vector , for any neighborhood of it, there exists a neighborhood of the scalar and neighborhood of , s.t.

**metric vector space** a topological space whose
topology is the topology of a metric/pseudometric space.

For metric vector space, continuity of vector + and scalar * can be expressed using sequences: is a metric vector space if:

**norm & seminorm** Given vector space
over field
. A function
is a seminorm on
if:

If in addition, , then it is a norm.

**Banach space**

a normed space that is a complete metric space.

Note that normed space is easily seen to be Hausdorff.

We can show that normed and seminormed spaces are metric spaces via defining . This requires proof because we are not just showing it is a metric space (this just requires verifying is a metric. We are actually showing it is a “metric vector space”, so we need to verify that + and scalar* are continuous operations.

**normed subspace** (of normed space
) Take
, and restrict the norm on
to on
.

Abstract/genreal normed spaces will be studied in Ch4. This chapter deals with special cases of normed spaces.

### Finite-Dim Spaces

. Equip this with the -norm for .

If , the norm is . If , we just have .

If , then the “norm” function defined above is not a norm, but we can still define a metric

can show that all the
spaces so defined, for
, are **complete & separable metric
spaces**.

(background info for isomorphic topological spaces)
**isomorphism (mapping) of vector spaces** source
Two vector spaces
over the same field are called **isomorphic**
if there is a bijection
which **preserves addition and scalar
multiplication**:

And this correspondence
is called **isomorphism of vector
spaces**.

**Isomorphic topological spaces** Topological vector
spaces
are called isomorphic if there exists a bijection
s.t.:

- is an isomorphism of vector spaces
- is a homeomorphism of topological spaces

Can show that , the spaces are all isomorphic. As a matter of fact, in Ch4, we will show that every two -dim normed spaces are isomorphic.

### Sequence spaces

for , elements in are defined to be sequences s.t.