The motivation behind the Hahn-Banach theorem can come across to a functional analysis newbie as somewhat elusive. I shall here try to explain this to the extent that I understand it.
Suppose that a seminorm on a vector space is such that iff , for some subspace of . Let be the dual space of . If is bounded with respect to this seminorm, then . We wish to induce via this seminorm a norm on the quotient space . Since a norm induces a metric (and a seminorm induces a pseudometric), it is natural then to define the norm on to be the distance corresponding to the seminorm between and , which is formally . It is easy to verify that this is well defined and a norm.
Similarly, induces an element of . Since is arbitrary, is also an arbitrary functional on its domain. We wish to show that given the constraint that for some subspace of , , we can for any , define functional on such that .
We are interested in extending a functional defined on a subspace to the full space with agreement of values on the subspace and a certain degree of boundnesses, more specifically an upper bound by the norm. Since this is trivially obtained by simply mapping to the elements outside the subspace, we are interested in an extension that is as non-zero or as large in absolute value as possible. In attempt to achieve this, we can try extending the functional with the requirement that its value on any input , which can be negative, is bounded above by the value of the application of another function on that has the reals, including negative ones, as its codomain. With this along with the properties of norms and seminorms in mind, we define the following.
Definition 1 Let be a real vector space. A sublinear functional on is a map such that for all and ,
- Triangle inequality: .
- Non-negative homogeneity: .
We immediately notice that the constraints defining a sublinear functional are a subset of the constraints defining a seminorm or norm, which means that any seminorm or norm is necessarily a sublinear functional, which means that any proposition that holds for an arbitrary sublinear functional also holds for an arbitrary seminorm or norm.
Lemma 1 We extend a linear functional , where is a subspace of , defined such that for all , for an arbitrary sublinear functional to the subspace such that for all .
Proof: must of course also be nonzero. Let be our extended functional, with and . We require that for any , for any ,
The case of is trivial.
In the case of , the inequality in is equivalent to
In the case of , the inequality in is equivalent to
Here, we notice that the product of any by any scalar is also in , by the closure property of subspace. Thus, if we show that for arbitrary ,
we have shown the existence of the desired . follows from
in which we used linearity of , the fact that on , and the triangle inequality on . This completes our proof.
The proof of the Hahn-Banach theorem, of which Lemma 1, is the most difficulty part, can come across as coming out of the blue. I certainly developed a better idea of how to derive it by “working backwards”, as done above, first assuming the existence of the desired property, then finding a condition that implies it, and finally proving that that condition is indeed satisfied.
Theorem 1 (Hahn Banach theorem) Let be a real vector space on which is defined a sublinear functional . Let be any subspace of it and be some linear functional such that for all . Then, there exists a linear functional such that for all , and for all , .
Proof: Lemma 1 tells us that if , we can always extend onto some subspace , which is a proper extension of subspace such the extension of is bounded above by on and agrees with on . Let be the collection of two-tuples such that is a linear functional defined on and bounded above by on . Let . We say that iff and on . One easy verifies that this is a partial order on . For any chain in , we take the union of all sets in the chain and define a function with for any , for some in the chain that is defined on a domain that contains . It is apparent that for any in the chain, . Thus, we can apply Zorn’s lemma to derive the existence of a maximal element in with respect to this partial order. The set associated with any maximal element must be itself in order for Lemma 1 to not be violated.
Now, we will go about generalized the Hahn-Banach theorem to complex vector spaces.
Lemma 2 Let be a complex vector space and let be a linear functional on . If is a complex linear functional on and , then is a real linear functional, and for all . Conversely, if is a real linear functional on and is defined by , then is complex linear. In this case, if is normed, we have .
Proof: Let Then, for any , we write , where . We have and . Thus, . For any , . That for any , is also easily verified.
For the converse, one easily verifies that , and for ,
For any , we have that . This shows that . With for some , , since is linear, we have that . This shows that for any , there exists a of the same norm such that , which shows that . This completes our proof.
In the proof of the above lemma, we omitted the case of .  introduced the notation
Using this we can define the polar decomposition of any as
Applying to gives us . We note that in the proof of Lemma 2, we multiplied by .
Lemma 3 For any complex vector space , there exists a real vector space and a function that is bijective and linear with respect to real but not complex coefficients.
Proof: For any , we must have and also for all . We also stipulate that for any , .Take any basis of . Then we have as a set of basis elements defining , . To verify that no non-trivial linear combination of basis elements of can equal , one can simply use linearity to derive violation of the definition of basis of in the case of linear dependence of a subset of basis elements of .
We have defined to be linear with respect to real coefficients. It is not at all linear with respect to complex coefficients as and are basis elements of a real vector space; since the underlying field is multiplying a vector of it by an imaginary number is simply not defined here.
Theorem 2 (Complex Hahn-Banach Theorem) Let be a complex vector space, a seminorm on , a subspace of , and a complex linear functional on such that for . Then there exists a complex linear functional on such that for all and
Proof: Let . By Lemma 3, there exists a real vector space with linear with respect to real coefficients and bijective. Let be defined by . One easily verifies that is real linear functional on . Moreover, on , which is a subspace of , , with a seminorm on easily verified as well. By the Hahn-Banach theorem for real vector spaces (Theorem 1), there is a real linear functional defined on that agrees with on such that for all , . From this we also derive an analogous extension of to , which is . Now, let on . As in the proof of Lemma 2, if , we have . Since on , we also have on . This completes our proof.
Theorem 3 Let be a normed vector space (over ). Then,
- If is a subspace of and , there exists such that and . In fact, if , can be defined such that , in which case .
- If , there exists such that and .
- The bounded linear functionals on separate points.
Proof: We wish to define on , which is, by linearity, done simply by prescribing the value of . The function on is such that if and only if . We need to prescribe an and we try the largest value which satisfies the requirement as given in the Hahn-Banach theorem, where the take to be the upper bounding seminorm. We let . Take arbitrary . We now wish to show that
The inequality in this is equivalent to
which is true from the definition of .
We have by the definition of operator norm that
Set and pick a sequence such that from above which exists by definition of . We have thus shown that . Obviously, from this we also have . Applying Hahn-Banach theorem to with as the seminorm to the extension of shows that we can extend the extension of all of , which completes the proof of (1).
(2) is a special case of (1) with . As for (3), if , ether is a complex multiple of or not. If yes, would suffice. If not, we can define on to be the constant zero function, and then, by (1), we can extend to a function on the entire space such that .
-  Gerald B. Folland. Real Analysis – Modern Techniques and their Applications. John Wiley & Sons, Inc., 1999.