The Banach Algebra of Borel Measures on Euclidean Space

This blog post is intended to deliver a quick explanation of the algebra of Borel measures on \(\mathbb{R}^n\). It will be broken into pieces. All complex-valued complex Borel measures \(M(\mathbb{R}^n)\) clearly form a vector space over \(\mathbb{C}\). The main goal of this post is to show that this is a Banach space and also a Banach algebra.

In fact the \(\mathbb{R}^n\) case can be generalised into any locally compact abelian group (see any abstract harmonic analysis books), this is because what really matters here is being locally compact and abelian. But at this moment we stick to Euclidean spaces. Note since \(\mathbb{R}^n\) is \(\sigma\)-compact, all Borel measures are regular.

To read this post you need to be familiar with some basic properties of Banach algebra, complex Borel measures, and the most important, Fubini's theorem.

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Left Shift Semigroup and Its Infinitesimal Generator

Left shift operator

Throughout we consider the Hilbert space \(L^2=L^2(\mathbb{R})\), the space of all complex-valued functions with real variable such that \(f \in L^2\) if and only if \[ \lVert f \rVert_2^2=\int_{-\infty}^{\infty}|f(t)|^2dm(t)<\infty \] where \(m\) denotes the ordinary Lebesgue measure (in fact it's legitimate to consider Riemann integral in this context).

For each \(t \geq 0\), we assign an bounded linear operator \(Q(t)\) such that \[ (Q(t)f)(s)=f(s+t). \] This is indeed bounded since we have \(\lVert Q(t)f \rVert_2 = \lVert f \rVert_2\) as the Lebesgue measure is translate-invariant. This is a left translation operator with a single step \(t\).

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Quasi-analytic Vectors and Hamburger Moment Problem (Operator Theory)

Analytic and quasi-analytic vectors

Guided by researches in function theory, operator theorists gave the analogue to quasi-analytic classes. Let \(A\) be an operator in a Banach space \(X\). \(A\) is not necessarily bounded hence the domain \(D(A)\) is not necessarily to be the whole space. We say \(x \in X\) is a \(C^\infty\) vector if \(x \in \bigcap_{n \geq 1}D(A^n)\). This is quite intuitive if we consider the differential operator. A vector is analytic if the series \[ \sum_{n=0}^{\infty}\lVert{A^n x}\rVert\frac{t^n}{n!} \] has a positive radius of convergence. Finally, we say \(x\) is quasi-analytic for \(A\) provided that \[ \sum_{n=0}^{\infty}\left(\frac{1}{\lVert A^n x \rVert}\right)^{1/n} = \infty \] or equivalently its nondecreasing majorant. Interestingly, if \(A\) is symmetric, then \(\lVert{A^nx}\rVert\) is log convex.

Based on the density of quasi-analytic vectors, we have an interesting result.

(Theorem) Let \(A\) be a symmetric operator in a Hilbert space \(\mathscr{H}\). If the set of quasi-analytic vectors spans a dense subset, then \(A\) is essentially self-adjoint.

This theorem can be considered as a corollary to the fundamental theorem of quasi-analytic classes, by applying suitable Banach space techniques in lieu.

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Several ways to prove Hardy's inequality

Suppose \(1 < p < \infty\) and \(f \in L^p((0,\infty))\) (with respect to Lebesgue measure of course) is a nonnegative function, take \[ F(x) = \frac{1}{x}\int_0^x f(t)dt \quad 0 < x <\infty, \] we have Hardy's inequality \(\def\lrVert[#1]{\lVert #1 \rVert}\) \[ \lrVert[F]_p \leq q\lrVert[f]_p \] where \(\frac{1}{p}+\frac{1}{q}=1\) of course.

There are several ways to prove it. I think there are several good reasons to write them down thoroughly since that may be why you find this page. Maybe you are burnt out since it's left as exercise. You are assumed to have enough knowledge of Lebesgue measure and integration.

Minkowski's integral inequality

Let \(S_1,S_2 \subset \mathbb{R}\) be two measurable set, suppose \(F:S_1 \times S_2 \to \mathbb{R}\) is measurable, then \[ \left[\int_{S_2} \left\vert\int_{S_1}F(x,y)dx \right\vert^pdy\right]^{\frac{1}{p}} \leq \int_{S_1} \left[\int_{S_2} |F(x,y)|^p dy\right]^{\frac{1}{p}}dx. \] A proof can be found at here by turning to Example A9. You may need to replace all measures with Lebesgue measure \(m\).

Now let's get into it. For a measurable function in this place we should have \(G(x,t)=\frac{f(t)}{x}\). If we put this function inside this inequality, we see \[ \begin{aligned} \lrVert[F]_p &= \left[\int_0^\infty \left\vert \int_0^x \frac{f(t)}{x}dt \right\vert^p dx\right]^{\frac{1}{p}} \\ &= \left[\int_0^\infty \left\vert \int_0^1 f(ux)du \right\vert^p dx\right]^{\frac{1}{p}} \\ &\leq \int_0^1 \left[\int_0^\infty |f(ux)|^pdx\right]^{\frac{1}{p}}du \\ &= \int_0^1 \left[\int_0^\infty |f(ux)|^pudx\right]^{\frac{1}{p}}u^{-\frac{1}{p}}du \\ &= \lrVert[f]_p \int_0^1 u^{-\frac{1}{p}}du \\ &=q\lrVert[f]_p. \end{aligned} \] Note we have used change-of-variable twice and the inequality once.

A constructive approach

I have no idea how people came up with this solution. Take \(xF(x)=\int_0^x f(t)t^{u}t^{-u}dt\) where \(0<u<1-\frac{1}{p}\). Hölder's inequality gives us \[ \begin{aligned} xF(x) &= \int_0^x f(t)t^ut^{-u}dt \\ &\leq \left[\int_0^x t^{-uq}dt\right]^{\frac{1}{q}}\left[\int_0^xf(t)^pt^{up}dt\right]^{\frac{1}{p}} \\ &=\left(\frac{1}{1-uq}x^{1-uq}\right)^{\frac{1}{q}}\left[\int_0^xf(t)^pt^{up}dt\right]^{\frac{1}{p}} \end{aligned} \] Hence \[ \begin{aligned} F(x)^p & \leq \frac{1}{x^p}\left\{\left(\frac{1}{1-uq}x^{1-uq}\right)^{\frac{1}{q}}\left[\int_0^xf(t)^pt^{up}dt\right]^{\frac{1}{p}}\right\}^{p} \\ &= \left(\frac{1}{1-uq}\right)^{\frac{p}{q}}x^{\frac{p}{q}(1-uq)-p}\int_0^x f(t)^pt^{up}dt \\ &= \left(\frac{1}{1-uq}\right)^{p-1}x^{-up-1}\int_0^x f(t)^pt^{up}dt \end{aligned} \]

Note we have used the fact that \(\frac{1}{p}+\frac{1}{q}=1 \implies p+q=pq\) and \(\frac{p}{q}=p-1\). Fubini's theorem gives us the final answer: \[ \begin{aligned} \int_0^\infty F(x)^pdx &\leq \int_0^\infty\left[\left(\frac{1}{1-uq}\right)^{p-1}x^{-up-1}\int_0^x f(t)^pt^{up}dt\right]dx \\ &=\left(\frac{1}{1-uq}\right)^{p-1}\int_0^\infty dx\int_0^x f(t)^pt^{up}x^{-up-1}dt \\ &=\left(\frac{1}{1-uq}\right)^{p-1}\int_0^\infty dt\int_t^\infty f(t)^pt^{up}x^{-up-1}dx \\ &=\left(\frac{1}{1-uq}\right)^{p-1}\frac{1}{up}\int_0^\infty f(t)^pdt. \end{aligned} \] It remains to find the minimum of \(\varphi(u) = \left(\frac{1}{1-uq}\right)^{p-1}\frac{1}{up}\). This is an elementary calculus problem. By taking its derivative, we see when \(u=\frac{1}{pq}<1-\frac{1}{p}\) it attains its minimum \(\left(\frac{p}{p-1}\right)^p=q^p\). Hence we get \[ \int_0^\infty F(x)^pdx \leq q^p\int_0^\infty f(t)^pdt, \] which is exactly what we want. Note the constant \(q\) cannot be replaced with a smaller one. We simply proved the case when \(f \geq 0\). For the general case, one simply needs to take absolute value.

Integration by parts

This approach makes use of properties of \(L^p\) space. Still we assume that \(f \geq 0\) but we also assume \(f \in C_c((0,\infty))\), that is, \(f\) is continuous and has compact support. Hence \(F\) is differentiable in this situation. Integration by parts gives \[ \int_0^\infty F^p(x)dx=xF(x)^p\vert_0^\infty- p\int_0^\infty xdF^p = -p\int_0^\infty xF^{p-1}(x)F'(x)dx. \] Note since \(f\) has compact support, there are some \([a,b]\) such that \(f >0\) only if \(0 < a \leq x \leq b < \infty\) and hence \(xF(x)^p\vert_0^\infty=0\). Next it is natural to take a look at \(F'(x)\). Note we have \[ F'(x) = \frac{f(x)}{x}-\frac{\int_0^x f(t)dt}{x^2}, \] hence \(xF'(x)=f(x)-F(x)\). A substitution gives us \[ \int_0^\infty F^p(x)dx = -p\int_0^\infty F^{p-1}(x)[f(x)-F(x)]dx, \] which is equivalent to say \[ \int_0^\infty F^p(x)dx = \frac{p}{p-1}\int_0^\infty F^{p-1}(x)f(x)dx. \] Hölder's inequality gives us \[ \begin{aligned} \int_0^\infty F^{p-1}(x)f(x)dx &\leq \left[\int_0^\infty F^{(p-1)q}(x)dx\right]^{\frac{1}{q}}\left[\int_0^\infty f(x)^pdx\right]^{\frac{1}{p}} \\ &=\left[\int_0^\infty F^{p}(x)dx\right]^{\frac{1}{q}}\left[\int_0^\infty f(x)^pdx\right]^{\frac{1}{p}}. \end{aligned} \] Together with the identity above we get \[ \int_0^\infty F^p(x)dx = q\left[\int_0^\infty F^{p}(x)dx\right]^{\frac{1}{q}}\left[\int_0^\infty f(x)^pdx\right]^{\frac{1}{p}} \] which is exactly what we want since \(1-\frac{1}{q}=\frac{1}{p}\) and all we need to do is divide \(\left[\int_0^\infty F^pdx\right]^{1/q}\) on both sides. So what's next? Note \(C_c((0,\infty))\) is dense in \(L^p((0,\infty))\). For any \(f \in L^p((0,\infty))\), we can take a sequence of functions \(f_n \in C_c((0,\infty))\) such that \(f_n \to f\) with respect to \(L^p\)-norm. Taking \(F=\frac{1}{x}\int_0^x f(t)dt\) and \(F_n = \frac{1}{x}\int_0^x f_n(t)dt\), we need to show that \(F_n \to F\) pointwise, so that we can use Fatou's lemma. For \(\varepsilon>0\), there exists some \(m\) such that \(\lrVert[f_n-f]_p < \frac{1}{n}\). Thus \[ \begin{aligned} |F_n(x)-F(x)| &= \frac{1}{x}\left\vert \int_0^x f_n(t)dt - \int_0^x f(t)dt \right\vert \\ &\leq \frac{1}{x} \int_0^x |f_n(t)-f(t)|dt \\ &\leq \frac{1}{x} \left[\int_0^x|f_n(t)-f(t)|^pdt\right]^{\frac{1}{p}}\left[\int_0^x 1^qdt\right]^{\frac{1}{q}} \\ &=\frac{1}{x^{1/p}}\left[\int_0^x|f_n(t)-f(t)|^pdt\right]^{\frac{1}{p}} \\ &\leq \frac{1}{x^{1/p}}\lrVert[f_n-f]_p <\frac{\varepsilon}{x^{1/p}}. \end{aligned} \] Hence \(F_n \to F\) pointwise, which also implies that \(|F_n|^p \to |F|^p\) pointwise. For \(|F_n|\) we have \[ \begin{aligned} \int_0^\infty |F_n(x)|^pdx &= \int_0^\infty \left\vert\frac{1}{x}\int_0^x f_n(t)dt\right\vert^p dx \\ &\leq \int_0^\infty \left[\frac{1}{x}\int_0^x |f_n(t)|dt\right]^{p}dx \\ &\leq q\int_0^\infty |f_n(t)|^pdt \end{aligned} \] note the third inequality follows since we have already proved it for \(f \geq 0\). By Fatou's lemma, we have \[ \begin{aligned} \int_0^\infty |F(x)|^pdx &= \int_0^\infty \lim_{n \to \infty}|F_n(x)|^pdx \\ &\leq \lim_{n \to \infty} \int_0^\infty |F_n(x)|^pdx \\ &\leq \lim_{n \to \infty}q^p\int_0^\infty |f_n(x)|^pdx \\ &=q^p\int_0^\infty |f(x)|^pdx. \end{aligned} \]

A Continuous Function Sending L^p Functions to L^1

Throughout, let \((X,\mathfrak{M},\mu)\) be a measure space where \(\mu\) is positive.

The question

If \(f\) is of \(L^p(\mu)\), which means \(\lVert f \rVert_p=\left(\int_X |f|^p d\mu\right)^{1/p}<\infty\), or equivalently \(\int_X |f|^p d\mu<\infty\), then we may say \(|f|^p\) is of \(L^1(\mu)\). In other words, we have a function \[ \begin{aligned} \lambda: L^p(\mu) &\to L^1(\mu) \\ f &\mapsto |f|^p. \end{aligned} \] This function does not have to be one to one due to absolute value. But we hope this function to be fine enough, at the very least, we hope it is continuous.

Here, \(f \sim g\) means that \(f-g\) equals to \(0\) almost everywhere with respect to \(\mu\). It can be easily verified that this is a equivalence relation.

Continuity

We still use \(\varepsilon-\delta\) argument but it's in a metric space. Suppose \((X,d_1)\) and \((Y,d_2)\) are two metric spaces and \(f:X \to Y\) is a function. We say \(f\) is continuous at \(x_0 \in X\) if for any \(\varepsilon>0\), there exists some \(\delta>0\) such that \(d_2(f(x_0),f(x))<\varepsilon\) whenever \(d_1(x_0,x)<\delta\). Further, we say \(f\) is continuous on \(X\) if \(f\) is continuous at every point \(x \in X\).

Metrics

For \(1\leq p<\infty\), we already have a metric by \[ d(f,g)=\lVert f-g \rVert_p \] given that \(d(f,g)=0\) if and only if \(f \sim g\). This is complete and makes \(L^p\) a Banach space. But for \(0<p<1\) (yes we are going to cover that), things are much more different, and there is one reason: Minkowski inequality holds reversely! In fact we have \[ \lVert f+g \rVert_p \geq \lVert f \rVert_p + \lVert g \rVert_p \] for \(0<p<1\). In fact, \(L^p\) space has too many weird things when \(0<p<1\). Precisely,

For \(0<p<1\), \(L^p(\mu)\) is locally convex if and only if \(\mu\) assumes finitely many values. (Proof.)

On the other hand, for example, \(X=[0,1]\) and \(\mu=m\) be the Lebesgue measure, then \(L^p(\mu)\) has no open convex subset other than \(\varnothing\) and \(L^p(\mu)\) itself. However,

A topological vector space \(X\) is normable if and only if its origin has a convex bounded neighbourhood. (See Kolmogorov's normability criterion.)

Therefore \(L^p(m)\) is not normable, hence not Banach.

We have gone too far. We need a metric that is fine enough.

Metric of \(L^p\) when \(0<p<1\)

In this subsection we always have \(0<p<1\).

Define \[ \Delta(f)=\int_X |f|^p d\mu \] for \(f \in L^p(\mu)\). We will show that we have a metric by \[ d(f,g)=\Delta(f-g). \] Fix \(y\geq 0\), consider the function \[ f(x)=(x+y)^p-x^p. \] We have \(f(0)=y^p\) and \[ f'(x)=p(x+y)^{p-1}-px^{p-1} \leq px^{p-1}-px^{p-1}=0 \] when \(x > 0\) and hence \(f(x)\) is nonincreasing on \([0,\infty)\), which implies that \[ (x+y)^p \leq x^p+y^p. \] Hence for any \(f\), \(g \in L^p\), we have \[ \Delta(f+g)=\int_X |f+g|^p d\mu \leq \int_X |f|^p d\mu + \int_X |g|^p d\mu=\Delta(f)+\Delta(g). \] This inequality ensures that \[ d(f,g)=\Delta(f-g) \] is a metric. It's immediate that \(d(f,g)=d(g,f) \geq 0\) for all \(f\), \(g \in L^p(\mu)\). For the triangle inequality, note that \[ d(f,h)+d(g,h)=\Delta(f-h)+\Delta(h-g) \geq \Delta((f-h)+(h-g))=\Delta(f-g)=d(f,g). \] This is translate-invariant as well since \[ d(f+h,g+h)=\Delta(f+h-g-h)=\Delta(f-g)=d(f,g) \] The completeness can be verified in the same way as the case when \(p>1\). In fact, this metric makes \(L^p\) a locally bounded F-space.

The continuity of \(\lambda\)

The metric of \(L^1\) is defined by \[ d_1(f,g)=\lVert f-g \rVert_1=\int_X |f-g|d\mu. \] We need to find a relation between \(d_p(f,g)\) and \(d_1(\lambda(f),\lambda(g))\), where \(d_p\) is the metric of the corresponding \(L^p\) space.

\(0<p<1\)

As we have proved, \[ (x+y)^p \leq x^p+y^p. \] Without loss of generality we assume \(x \geq y\) and therefore \[ x^p=(x-y+y)^p \leq (x-y)^p+y^p. \] Hence \[ x^p-y^p \leq (x-y)^p. \] By interchanging \(x\) and \(y\), we get \[ |x^p-y^p| \leq |x-y|^p. \] Replacing \(x\) and \(y\) with \(|f|\) and \(|g|\) where \(f\), \(g \in L^p\), we get \[ \int_{X}\lvert |f|^p-|g|^p \rvert d\mu \leq \int_X |f-g|^p d\mu. \] But \[ d_1(\lambda(f),\lambda(g))=\int_{X}\lvert |f|^p-|g|^p \rvert d\mu \\ d_p(f,g)=\Delta(f-g)= d\mu \leq \int_X |f-g|^p d\mu \] and we therefore have \[ d_1(\lambda(f),\lambda(g)) \leq d_p(f,g). \] Hence \(\lambda\) is continuous (and in fact, Lipschitz continuous and uniformly continuous) when \(0<p<1\).

\(1 \leq p < \infty\)

It's natural to think about Minkowski's inequality and Hölder's inequality in this case since they are critical inequality enablers. You need to think about some examples of how to create the condition to use them and get a fine result. In this section we need to prove that \[ |x^p-y^p| \leq p|x-y|(x^{p-1}+y^{p-1}). \] This inequality is surprisingly easy to prove however. We will use nothing but the mean value theorem. Without loss of generality we assume that \(x > y \geq 0\) and define \(f(t)=t^p\). Then \[ \frac{f(x)-f(y)}{x-y}=f'(\zeta)=p\zeta^{p-1} \] where \(y < \zeta < x\). But since \(p-1 \geq 0\), we see \(\zeta^{p-1} < x^{p-1} <x^{p-1}+y^{p-1}\). Therefore \[ f(x)-f(y)=x^p-y^p=p(x-y)\zeta^{p-1}<p(x-y)(x^{p-1}-y^{p-1}). \] For \(x=y\) the equality holds.


Therefore \[ \begin{aligned} d_1(\lambda(f),\lambda(g)) &= \int_X \left||f|^p-|g|^p\right|d\mu \\ &\leq \int_Xp\left||f|-|g|\right|(|f|^{p-1}+|g|^{p-1})d\mu \end{aligned} \] By Hölder's inequality, we have \[ \begin{aligned} \int_X ||f|-|g||(|f|^{p-1}+|g|^{p-1})d\mu & \leq \left[\int_X \left||f|-|g|\right|^pd\mu\right]^{1/p}\left[\int_X\left(|f|^{p-1}+|g|^{p-1}\right)^q\right]^{1/q} \\ &\leq \left[\int_X \left|f-g\right|^pd\mu\right]^{1/p}\left[\int_X\left(|f|^{p-1}+|g|^{p-1}\right)^q\right]^{1/q} \\ &=\lVert f-g \rVert_p \left[\int_X\left(|f|^{p-1}+|g|^{p-1}\right)^q\right]^{1/q}. \end{aligned} \] By Minkowski's inequality, we have \[ \left[\int_X\left(|f|^{p-1}+|g|^{p-1}\right)^q\right]^{1/q} \leq \left[\int_X|f|^{(p-1)q}d\mu\right]^{1/q}+\left[\int_X |g|^{(p-1)q}d\mu\right]^{1/q} \] Now things are clear. Since \(1/p+1/q=1\), or equivalently \(1/q=(p-1)/p\), suppose \(\lVert f \rVert_p\), \(\lVert g \rVert_p \leq R\), then \((p-1)q=p\) and therefore \[ \left[\int_X|f|^{(p-1)q}d\mu\right]^{1/q}+\left[\int_X |g|^{(p-1)q}d\mu\right]^{1/q} = \lVert f \rVert_p^{p-1}+\lVert g \rVert_p^{p-1} \leq 2R^{p-1}. \] Summing the inequalities above, we get \[ \begin{aligned} d_1(\lambda(f),\lambda(g)) \leq 2pR^{p-1}\lVert f-g \rVert_p =2pR^{p-1}d_p(f,g) \end{aligned} \] hence \(\lambda\) is continuous.

Conclusion and further

We have proved that \(\lambda\) is continuous, and when \(0<p<1\), we have seen that \(\lambda\) is Lipschitz continuous. It's natural to think about its differentiability afterwards, but the absolute value function is not even differentiable so we may have no chance. But this is still a fine enough result. For example we have no restriction to \((X,\mathfrak{M},\mu)\) other than the positivity of \(\mu\). Therefore we may take \(\mathbb{R}^n\) as the Lebesgue measure space here, or we can take something else.

It's also interesting how we use elementary Calculus to solve some much more abstract problems.

The Big Three Pt. 6 - Closed Graph Theorem with Applications

(Before everything: elementary background of topology and vector spaces, Banach spaces, is assumed.)

A surprising result of Banach spaces

We can define several relations between two norms. Suppose we have a topological vector space \(X\) and two norms \(\lVert \cdot \rVert_1\) and \(\lVert \cdot \rVert_2\). One says \(\lVert \cdot \rVert_1\) is weaker than \(\lVert \cdot \rVert_2\) if there is \(K>0\) such that \(\lVert x \rVert_1 \leq K \lVert x \rVert_2\) for all \(x \in X\). Two norms are equivalent if each is weaker than the other (trivially this is a equivalence relation). The idea of stronger and weaker norms is related to the idea of the "finer" and "coarser" topologies in the setting of topological spaces.

So what about their limit of convergence? Unsurprisingly this can be verified with elementary \(\epsilon-N\) arguments. Suppose now \(\lVert x_n - x \rVert_1 \to 0\) as \(n \to 0\), we immediately have \[ \lVert x_n - x \rVert_2 \leq K \lVert x_n-x \rVert_1 < K\varepsilon \]

for some large enough \(n\). Hence \(\lVert x_n - x \rVert_2 \to 0\) as well. But what about the converse? We give a new definition of equivalence relation between norms.

(Definition) Two norms \(\lVert \cdot \rVert_1\) and \(\lVert \cdot \rVert_2\) of a topological vector space are compatible if given that \(\lVert x_n - x \rVert_1 \to 0\) and \(\lVert x_n - y \rVert_2 \to 0\) as \(n \to \infty\), we have \(x=y\).

By the uniqueness of limit, we see if two norms are equivalent, then they are compatible. And surprisingly, with the help of the closed graph theorem we will discuss in this post, we have

(Theorem 1) If \(\lVert \cdot \rVert_1\) and \(\lVert \cdot \rVert_2\) are compatible, and both \((X,\lVert\cdot\rVert_1)\) and \((X,\lVert\cdot\rVert_2)\) are Banach, then \(\lVert\cdot\rVert_1\) and \(\lVert\cdot\rVert_2\) are equivalent.

This result looks natural but not seemingly easy to prove, since one find no way to build a bridge between the limit and a general inequality. But before that, we need to elaborate some terminologies.

Preliminaries

(Definition) For \(f:X \to Y\), the graph of \(f\) is defined by \[ G(f)=\{(x,f(x)) \in X \times Y:x \in X\}. \]

If both \(X\) and \(Y\) are topological spaces, and the topology of \(X \times Y\) is the usual one, that is, the smallest topology that contains all sets \(U \times V\) where \(U\) and \(V\) are open in \(X\) and \(Y\) respectively, and if \(f: X \to Y\) is continuous, it is natural to expect \(G(f)\) to be closed. For example, by taking \(f(x)=x\) and \(X=Y=\mathbb{R}\), one would expect the diagonal line of the plane to be closed.

(Definition) The topological space \((X,\tau)\) is an \(F\)-space if \(\tau\) is induced by a complete invariant metric \(d\). Here invariant means that \(d(x+z,y+z)=d(x,y)\) for all \(x,y,z \in X\).

A Banach space is easily to be verified to be a \(F\)-space by defining \(d(x,y)=\lVert x-y \rVert\).

(Open mapping theorem) See this post

By definition of closed set, we have a practical criterion on whether \(G(f)\) is closed.

(Proposition 1) \(G(f)\) is closed if and only if, for any sequence \((x_n)\) such that the limits \[ x=\lim_{n \to \infty}x_n \quad \text{ and }\quad y=\lim_{n \to \infty}f(x_n) \] exist, we have \(y=f(x)\).

In this case, we say \(f\) is closed. For continuous functions, things are trivial.

(Proposition 2) If \(X\) and \(Y\) are two topological spaces and \(Y\) is Hausdorff, and \(f:X \to Y\) is continuous, then \(G(f)\) is closed.

Proof. Let \(G^c\) be the complement of \(G(f)\) with respect to \(X \times Y\). Fix \((x_0,y_0) \in G^c\), we see \(y_0 \neq f(x_0)\). By the Hausdorff property of \(Y\), there exists some open subsets \(U \subset Y\) and \(V \subset Y\) such that \(y_0 \in U\) and \(f(x_0) \in V\) and \(U \cap V = \varnothing\). Since \(f\) is continuous, we see \(W=f^{-1}(V)\) is open in \(X\). We obtained a open neighborhood \(W \times U\) containing \((x_0,y_0)\) which has empty intersection with \(G(f)\). This is to say, every point of \(G^c\) has a open neighborhood contained in \(G^c\), hence a interior point. Therefore \(G^c\) is open, which is to say that \(G(f)\) is closed. \(\square\)

closed-graph

REMARKS. For \(X \times Y=\mathbb{R} \times \mathbb{R}\), we have a simple visualization. For \(\varepsilon>0\), there exists some \(\delta\) such that \(|f(x)-f(x_0)|<\varepsilon\) whenever \(|x-x_0|<\delta\). For \(y_0 \neq f(x_0)\), pick \(\varepsilon\) such that \(0<\varepsilon<\frac{1}{2}|f(x_0)-y_0|\), we have two boxes (\(CDEF\) and \(GHJI\) on the picture), namely \[ B_1=\{(x,y):x_0-\delta<x<x_0+\delta,f(x_0)-\varepsilon<y<f(x_0)+\varepsilon\} \] and \[ B_2=\{(x,y):x_0-\delta<x<x_0+\delta,y_0-\varepsilon<y<y_0+\varepsilon\}. \] In this case, \(B_2\) will not intersect the graph of \(f\), hence \((x_0,y_0)\) is an interior point of \(G^c\).

The Hausdorff property of \(Y\) is not removable. To see this, since \(X\) has no restriction, it suffices to take a look at \(X \times X\). Let \(f\) be the identity map (which is continuous), we see the graph \[ G(f)=\{(x,x):x \in X\} \] is the diagonal. Suppose \(X\) is not Hausdorff, we reach a contradiction. By definition, there exists some distinct \(x\) and \(y\) such that all neighborhoods of \(x\) contain \(y\). Pick \((x,y) \in G^c\), then all neighborhoods of \((x,y) \in X \times X\) contain \((x,x)\) so \((x,y) \in G^c\) is not a interior point of \(G^c\), hence \(G^c\) is not open.

Also, as an immediate consequence, every affine algebraic variety in \(\mathbb{C}^n\) and \(\mathbb{R}^n\) is closed with respect to Euclidean topology. Further, we have the Zariski topology \(\mathcal{Z}\) by claiming that, if \(V\) is an affine algebraic variety, then \(V^c \in \mathcal{Z}\). It's worth noting that \(\mathcal{Z}\) is not Hausdorff (example?) and in fact much coarser than the Euclidean topology although an affine algebraic variety is both closed in the Zariski topology and the Euclidean topology.

The closed graph theorem

After we have proved this theorem, we are able to prove the theorem about compatible norms. We shall assume that both \(X\) and \(Y\) are \(F\)-spaces, since the norm plays no critical role here. This offers a greater variety but shall not be considered as an abuse of abstraction.

(The Closed Graph Theorem) Suppose

  1. \(X\) and \(Y\) are \(F\)-spaces,

  2. \(f:X \to Y\) is linear,

  3. \(G(f)\) is closed in \(X \times Y\).

Then \(f\) is continuous.

In short, the closed graph theorem gives a sufficient condition to claim the continuity of \(f\) (keep in mind, linearity does not imply continuity). If \(f:X \to Y\) is continuous, then \(G(f)\) is closed; if \(G(f)\) is closed and \(f\) is linear, then \(f\) is continuous.

Proof. First of all we should make \(X \times Y\) an \(F\)-space by assigning addition, scalar multiplication and metric. Addition and scalar multiplication are defined componentwise in the nature of things: \[ \alpha(x_1,y_1)+\beta(x_2,y_2)=(\alpha x_1+\beta x_2,\alpha y_1 + \beta y_2). \] The metric can be defined without extra effort: \[ d((x_1,y_1),(x_2,y_2))=d_X(x_1,x_2)+d_Y(y_1,y_2). \] Then it can be verified that \(X \times Y\) is a topological space with translate invariant metric. (Potentially the verifications will be added in the future but it's recommended to do it yourself.)


Since \(f\) is linear, the graph \(G(f)\) is a subspace of \(X \times Y\). Next we quote an elementary result in point-set topology, a subset of a complete metric space is closed if and only if it's complete, by the translate-invariance of \(d\), we see \(G(f)\) is an \(F\)-space as well. Let \(p_1: X \times Y \to X\) and \(p_2: X \times Y \to Y\) be the natural projections respectively (for example, \(p_1(x,y)=x\)). Our proof is done by verifying the properties of \(p_1\) and \(p_2\) on \(G(f)\).

For simplicity one can simply define \(p_1\) on \(G(f)\) instead of the whole space \(X \times Y\), but we make it a global projection on purpose to emphasize the difference between global properties and local properties. One can also write \(p_1|_{G(f)}\) to dodge confusion.


Claim 1. \(p_1\) (with restriction on \(G(f)\)) defines an isomorphism between \(G(f)\) and \(X\).

For \(x \in X\), we see \(p_1(x,f(x)) = x\) (surjectivity). If \(p_1(x,f(x))=0\), we see \(x=0\) and therefore \((x,f(x))=(0,0)\), hence the restriction of \(p_1\) on \(G\) has trivial kernel (injectivity). Further, it's trivial that \(p_1\) is linear.

Claim 2. \(p_1\) is continuous on \(G(f)\).

For every sequence \((x_n)\) such that \(\lim_{n \to \infty}x_n=x\), we have \(\lim_{n \to \infty}f(x_n)=f(x)\) since \(G(f)\) is closed, and therefore \(\lim_{n \to \infty}p_1(x_n,f(x_n)) =x\). Meanwhile \(p_1(x,f(x))=x\). The continuity of \(p_1\) is proved.

Claim 3. \(p_1\) is a homeomorphism with restriction on \(G(f)\).

We already know that \(G(f)\) is an \(F\)-space, so is \(X\). For \(p_1\) we have \(p_1(G(f))=X\) is of the second category (since it's an \(F\)-space and \(p_1\) is one-to-one), and \(p_1\) is continuous and linear on \(G(f)\). By the open mapping theorem, \(p_1\) is an open mapping on \(G(f)\), hence is a homeomorphism thereafter.

Claim 4. \(p_2\) is continuous.

This follows the same way as the proof of claim 2 but much easier since we have no need to care about \(f\).


Now things are immediate once one realizes that \(f=p_2 \circ p_1|_{G(f)}^{-1}\), and hence \(f\) is continuous. \(\square\)

Applications

Before we go for theorem 1 at the beginning, we drop an application on Hilbert spaces.

Let \(T\) be a bounded operator on the Hilbert space \(L_2([0,1])\) so that if \(\phi \in L_2([0,1])\) is a continuous function so is \(T\phi\). Then the restriction of \(T\) to \(C([0,1])\) is a bounded operator of \(C([0,1])\).

For details please check this.


Now we go for the identification of norms. Define \[ \begin{aligned} f:(X,\lVert\cdot\rVert_1) &\to (X,\lVert\cdot\rVert_2) \\ x &\mapsto x \end{aligned} \] i.e. the identity map between two Banach spaces (hence \(F\)-spaces). Then \(f\) is linear. We need to prove that \(G(f)\) is closed. For the convergent sequence \((x_n)\) \[ \lim_{n \to \infty}\lVert x_n -x \rVert_1=0, \] we have \[ \lim_{n \to \infty} \lVert f(x_n)-x \rVert_2=\lim_{n \to \infty}\lVert x_n -x\rVert_2=\lim_{n \to \infty}\lVert f(x_n)-f(x)\rVert_2=0. \] Hence \(G(f)\) is closed. Therefore \(f\) is continuous, hence bounded, we have some \(K\) such that \[ \lVert x \rVert_2 =\lVert f(x) \rVert_1 \leq K \lVert x \rVert_1. \] By defining \[ \begin{aligned} g:(X,\lVert\cdot\rVert_2) &\to (X,\lVert\cdot\rVert_1) \\ x &\mapsto x \end{aligned} \] we see \(g\) is continuous as well, hence we have some \(K'\) such that \[ \lVert x \rVert_1 =\lVert g(x) \rVert_2 \leq K'\lVert x \rVert_2 \] Hence two norms are weaker than each other.

The series

Since there is no strong reason to write more posts on this topic, i.e. the three fundamental theorems of linear functional analysis, I think it's time to make a list of the series. It's been around half a year.

References

  • Walter Rudin, Functional Analysis
  • Peter Lax, Functional Analysis
  • Jesús Gil de Lamadrid, Some Simple Applications of the Closed Graph Theorem