AN INTRODUCTION TO BANACH AND HILBERT SPACES
FRANK LIOU
1. Some Important Function Spaces:
Let S be a nonempty set. The set of all real valued functions from S to R is denoted by F (S, R). For convenience, F(S, R) is denoted by F(S). For f, g ∈ F(S, R) and a ∈ R, we define the sum f + g : S → R of f and g and the scalar multiplication (af ) : S → R of f by a as follows. For s ∈ S,
(f + g)(s) = f (s) + g(s), (af )(s) = af (s).
The zero function 0 : S → R is defined to be 0(s) = 0 for all s ∈ S.
Proposition 1.1. The set F (S, R) forms a real vector space.
Proof. The proof is left to the reader.
Proposition 1.2. Let S be a finite set and N be the number of elements of S. Then F (S) is linear isomorphic to RN :
F (S) ∼= RN.
Proof. Assume that S contains N distinct element and denote S by {s1, · · · , sN}. For each f ∈ F (S), we define
T (f ) = (f (s1), · · · , f (sN)).
Let us check that T : F (S) → RN is a linear isomorphism. Checking the linearity of T is routine. Let us show that the map is bijective. For a = (a1, · · · , aN) ∈ RN, we define a function f : S → R by f (si) = ai. Then T (f ) = a. For f ∈ ker T, then f (s1) = · · · = f (sN) = 0. Thus f is the zero function. We find ker T = {0}.
When S = {1, · · · , N }, there is a canonical choice of linear isomorphism between F ({1, · · · , N }) and RN. For f ∈ F ({1, · · · , N }), we denote f (i) by fi. The canonical choice of linear iso- morphism T : F ({1, · · · , N }) → RN is defined by T (f ) = (f1, · · · , fN). Thus RN can be defined as the space of functions a : {1, · · · , N } → R.
Definition 1.1. A function f ∈ F (S) is said to be bounded if there exists M > 0 such that
|f (s)| ≤ M for all s ∈ S.
Let B(S) be the subset of F (S) consisting of bounded functions.
Proposition 1.3. The subset B(S) forms a real vector subspace of F (S).
Proof. Let f, g ∈ B(S). Choose two positive real numbers M1 and M2 such that
|f (s)| ≤ M1, |g(s)| ≤ M2 for all s ∈ S. By triangle inequality,
|f (s) + g(s)| ≤ |f (s)| + |g(s)| ≤ M1+ M2 for any s ∈ S.
1
Hence M1+ M2 is an upper bound for {|f (s) + g(s)| : s ∈ S}. Thus f + g : S → R is a bounded function, i.e. f + g ∈ B(S). Since |0(s)| = |0| ≤ 1 for any s ∈ S, 0 ∈ B(S). For any a ∈ R, |af (s)| ≤ |a|M1 for any s ∈ S, af ∈ B(S).
Definition 1.2. A norm on a real vector space V is a function
k · k : V → R such that
(1) kavk = |a|kvk for any a ∈ R and for any v ∈ V ; (2) kvk ≥ 0; kvk = 0 if and only if v = 0;
(3) kv + wk ≤ kvk + kwk for any v, w.
A real vector space together with a norm (V, k · k) is called a real normed (vector) space.
Lemma 1.1. For each f ∈ B(S), we set
kf k∞= sup
s∈S
|f (s)|.
The function k · k∞: B(S) → R defines a norm on B(S).
Proof. For a ∈ R,
kaf k∞= sup
s∈S
|af (s)| = sup
s∈S
|a||f (s)| = |a| sup
s∈S
|f (s)| = |a|kf k∞.
For any s ∈ S, |f (s)| ≥ 0 and hence kf k∞ ≥ 0. If kf k∞ = 0, then 0 ≤ |f (s)| ≤ sups∈S|f (s)| = 0 for any s ∈ S. Therefore |f (s)| = 0 for all s ∈ S. This implies that f is the zero function, i.e. f (s) = 0 for all s ∈ S. For any f, g ∈ B(S) and any s ∈ S,
|f (s) + g(s)| ≤ |f (s)| + |g(s)| ≤ kf k∞+ kgk∞.
We find that kf k∞+ kgk∞ is an upper bound of {|f (s) + g(s)| : s ∈ S}. By definition of
sup, kf + gk∞≤ kf k∞+ kgk∞.
Any norm k · k on a vector space V defines a metric d on V by d(v, w) = kv − wk.
This metric is called the metric induced from k · k.
Definition 1.3. A real normed space V is called a real Banach space if V together with the metric induced from the norm is a complete metric space. A real Banach space is a complete real normed space.
Lemma 1.2. Let (an) be a sequence of real numbers convergent to a ∈ R. Suppose that M > 0 is a real number such that |an| ≤ M for all n ≥ 1. Then |a| ≤ M.
Proof. Suppose not. Assume that |a| > M. Take = (|a| − M )/2. Since (an) is convergent to a in R, we can choose N ∈ N so that |an− a| < = |a|−M2 < |a| − M when n ≥ N. This implies that |an| ≥ |a| − |an− a| > M whenever n ≥ N. This leads to the contradiction to
our assumption.
Theorem 1.1. The normed space (B(S), k · k∞) is a real Banach space.
Proof. Let (fn) be a Cauchy sequence in B(S). For any > 0, there exists N ∈ N such that kfn− fmk∞< /2. whenever n, m ≥ N. For s ∈ S,
|fn(s) − fm(s)| ≤ kfn− fmk∞< 2
whenever n, m ≥ N. This shows that The sequence (fn(s)) of real numbers is a Cauchy sequence in R. By completeness of R, (fn(s)) is convergent in R. We denote the limit of (fn(s)) by f (s), i.e. f (s) = limn→∞fn(s). We obtain a function f : S → R. Let us prove that f ∈ B(S) and (fn) is convergent to f in B(S).
Since (fn) is a Cauchy sequence in B(S), it is bounded in B(S). There exists M > 0 so that kfnk∞≤ M for all n ≥ 1. For s ∈ S,
|fn(s)| ≤ kfnk∞≤ M.
We see that the sequence (fn(s)) of real numbers is bounded. By Lemma 1.2, we find
|f (s)| ≤ M for all s ∈ S. Thus f : S → R is a bounded function. We see that f ∈ B(S). By taking m → ∞ of |fn(s) − fm(s)| < /2, we find that for s ∈ S
|fn(s) − f (s)| ≤ 2
whenever n ≥ N. This shows that kfn− f k∞ ≤ /2 < when n ≥ N. This proves that
(fn) is convergent to f in B(S).
When S = {s1, · · · , sN} is a finite set, we can see that B(S) = F (S). In fact, for each f ∈ F (S), Im f = {f (s1), · · · , f (sN)} is a finite subset of R and hence Im f is bounded.
Moreover,
kf k∞= max{|f (s1)|, · · · , |f (sN)|}.
Using the isomorphism F (S) ∼= RN in Proposition 1.2, we can define the infinity norm on RN as follows. For each a = (a1, · · · , aN) in RN, we set
kak∞= max{|a1|, · · · , |aN|}.
We leave it to the reader to verify that (RN, k · k∞) is a real normed space. On RN, there are also other different norms. Let us recall that the Euclidean norm of a is given by kak = Pn
i=1|ai|21/2
. For each 1 ≤ p < ∞, define
kakp=
n
X
i=1
|ai|p
!1/p
.
Proposition 1.4. For 1 ≤ p ≤ ∞, (RN, k · kp) is a real Banach space. Moreover, kak∞= limp→∞kakp.
Proof. We leave it to the reader to verify that k · kp defines a norm on RN. See hw 7.
Let (an) be a Cauchy sequence in (RN, k · kp). For any > 0, there exists N ∈ N such that kan− amkp < /2√p
N for any n, m ≥ N. Denote an by (a1n, · · · , aNn) for n ≥ 1. For 1 ≤ i ≤ N,
(1.1) |ain− aim| ≤ kan− amkp< 2√p
N
whenever n, m ≥ N. This shows that (ain) is a Cauchy sequence in R for 1 ≤ i ≤ N. By completeness of R, (ain) are all convergent. Let ai be the limit of (ain) for all i. By taking the limit of (1.1) as m → ∞, we find
|ain− ai| ≤ 2√p
N <
√p
N
whenever n ≥ N. Let a = (a1, · · · , an) ∈ RN. Then
kan− akpp =
N
X
i=1
|ain− ai|p < p
N · N = p
whenever n ≥ N. Thus kan− akp < whenever n ≥ N. We prove that (an) is convergent
to a in (RN, k · kp).
When S = N, we call F(N) the space of sequences of real numbers. In this case, we denote the Banach space (B(N), k · k∞) by l∞(N). For 1 ≤ p < ∞. let lp(N) be the subset of F (N) consisting of sequences a : N → R such thatP∞
i=1|a(i)|p < ∞. For each a ∈ lp(N ), we define
kakp =
∞
X
i=1
|a(i)|p
!1/p
Lemma 1.3. For each 1 ≤ p < ∞, we obtain a normed space (lp(N), k · kp).
Proof. The proof basically follows from Holder inequality and the Minkowski inequality for infinite series and is almost the same as that of (RN, k · kp). We simply denote the norm space (lp(N), k · kp) by lp(N). In fact, we can prove further that:
Theorem 1.2. For 1 ≤ p ≤ ∞, the normed space lp(N) is a real Banach space.
Proof. The proof is very similar to that of the completeness of (RN, k · kp). The details are
left to the reader as an exercise. See hw 7.
Lemma 1.4. For a, b ∈ l2(N), the infinite seriesP∞
i=1a(i)b(i) is absolutely convergent.
For a, b ∈ l2(N), we define
ha, bi =
∞
X
i=1
a(i)b(i).
Then h·, ·i defines an inner product on l2(N).
Definition 1.4. Let V be a real vector space. An inner product on V is a function h·, ·i : V × V → R such that for any u, v, w ∈ V and a, b ∈ R,
(1) hau + bv, w·i = ahu, wi + bhv, wi, (2) hu, vi = hv, ui,
(3) hu, ui ≥ 0 and = 0 if and only if u = 0.
A real vector space V together with an inner product h·, ·i is called a real inner product space.
An inner product h·, ·i on V induces a norm on V defined by kvk =phv, vi for v ∈ V . We call k · k the norm induced from h·, ·i.
Definition 1.5. An inner product space (V, h·, ·i) is called a real Hilbert space if V together with the normed induced from h·, ·i is a real Banach space. A real Hilbert space is a complete inner product space.
For a ∈ l2(N), we see that
ha, ai = kak2. This implies that
Theorem 1.3. The inner product space (l2(N), h·, ·i) is a real Hilbert space.
We can also consider complex Banach spaces and complex Hilbert spaces. We leave the definitions to the readers.
