Seeing a Function as a Single Point — An Invitation to Infinite Dimensions
A Function as a Single Point
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Say "function," and most people picture a graph: put $x$ on the horizontal axis and $y$ on the vertical, and a single curve wriggles across the page. That picture is certainly correct. But functional analysis crumples this whole curve up and treats it as a single point floating in a space.
A curve becoming a point — heard for the first time, it may sound like a joke. Yet this seemingly reckless way of looking is exactly the powerful idea that runs from Fourier analysis all the way to quantum mechanics. Once we regard a curve as a point, the tools we have used in linear algebra — "length," "angle," "projection" — carry over to functions almost unchanged; that is the payoff of this view. In this piece, let us savor that idea with our shoulders relaxed.
Starting by counting coordinates
A point in the plane can be named by two numbers $(x, y)$; a point in space by three, $(x, y, z)$. A tuple of numbers pins down a single point — everyone knows this much. So what if we line up four numbers? Or five? Ten is, in principle, no problem at all. A tuple of $n$ numbers $(a_1, a_2, \dots, a_n)$ is a single point of $n$-dimensional space. Even if we cannot see it, it is a perfectly respectable point as far as calculation is concerned.
Here we get just a little bold. Let us keep stretching the tuple until at last we line up infinitely many numbers, $(a_1, a_2, a_3, \dots)$, and decide to think of this, too, as a single point. Then a natural question arises: could a function $f$ also be regarded as an "infinitely long tuple of numbers," namely its values $f(x_1), f(x_2), \dots$ at each point?
And indeed it can. A function is a rule that specifies countless values at once. If we regard those values as coordinates, then a single function becomes just one point in an infinite-dimensional space. That is what "crumpling the curve up" meant. (Strictly, since a function is defined on a continuum, it is closer to call it "an object holding infinitely many values at once" than "an infinitely long sequence of numbers.")
A function is also a vector
Once it looks like a point, addition comes next. Just as two vectors can be added, two functions can be added: $ (f+g)(x) = f(x) + g(x) $, adding the values at each point. Scalar multiplication is the same, $ (cf)(x) = c\,f(x) $. In other words, a function admits addition and scalar multiplication — it is a full-fledged vector. After all, a vector is anything that can be added and scaled by a constant; arrows are not the only vectors.
Once we know it is a vector, the tools we grew familiar with in finite-dimensional linear algebra transfer over intact. If we want to measure length, we bring in a "norm"; if we want to talk about angles and orthogonality, we bring in an "inner product." We move the geometry we know from $\mathbb{R}^n$ into the world of functions.
A quick note: the inner product gives birth to "orthogonal functions"
If the inner product of two vectors is zero, they are orthogonal. For functions, too, we say $f$ and $g$ are "orthogonal" when $\langle f, g \rangle = \int f(x)\,g(x)\,dx = 0$. For instance, $\sin x$ and $\cos x$ are indeed orthogonal when integrated over a suitable interval. Two functions meeting at a right angle — we cannot draw it, but the calculation follows our intuition faithfully.
Measuring "are two functions close?"
Now that they are points, we would like to measure the distance between them. How "close" are two functions $f$ and $g$? In fact, there is more than one way to measure it.
One way looks at the place where they differ the most and says "at worst they differ by this much" (uniform closeness). Another squares the difference, averages it over the whole domain, and says "on the whole they differ by this much" (the $L^2$ notion of closeness). For the very same pair of functions, changing the ruler can flip the verdict between "close" and "far." Only once we decide which ruler to measure with does the space of functions acquire a shape.
How to choose this "ruler = norm" is the first fork in the road of functional analysis. It is less that one is correct than that we pick the ruler that suits the problem — much like choosing what clothes to wear.
The delightful part of this viewpoint
Translate functions into points, and collections of functions into a space, and the scattered stories begin to connect on a single map.
Take Fourier series. That trick of expressing a complicated periodic function as a sum of $\sin nx$ and $\cos nx$ is nothing other than the operation of "taking mutually orthogonal functions as a basis and resolving the target point along those directions." In spirit it is exactly the same as splitting a vector in $\mathbb{R}^3$ along the $x, y, z$ axes — only now the basis has infinitely many elements.
Or consider least squares. Choosing the line or curve closest to the data is precisely the geometry of "dropping a perpendicular from a point onto a subspace." A hard approximation problem transforms in a flash into the familiar problem of "the foot of a perpendicular." Techniques devised separately in different fields can be retold in one geometric language — that is the reward for seeing functions as a space.
That said, infinity is a bit strange
Even so, infinite dimensions are not a docile extension of finite dimensions. The new home has some unfamiliar scenery.
A quick note: even bounded and closed, "gaps" can remain
In finite dimensions, from a bounded, closed set one could always extract a convergent subsequence from any sequence of points (points must accumulate somewhere). In infinite dimensions, however, if we line up countless mutually orthogonal functions, it can happen that the set is bounded yet no subsequence approaches any single point. "Pack them in tightly and still they do not accumulate" — a strangeness peculiar to infinite dimensions, where there are boundlessly many directions. That is exactly why completeness (the property that a sequence that ought to converge really does), which we learn later, becomes decisively important in infinite dimensions.
Closing — off to view things as points
A curve into a single point; a collection of functions into a single space. Just this switch of viewpoint lays the tools of analysis, geometry, and algebra out together on one desk. From a function as "something to draw" to "something to house" — this move is the entrance to the field called functional analysis. In other words, functional analysis is less about learning new computational techniques than about acquiring a new coordinate system, a new geometry, for seeing functions.
So let us bring the tools into that space one by one. A ruler (norm) and an angle (inner product), and the completeness unique to the infinite — in the basic level we assemble these carefully and build the comfortable dwellings called Banach spaces and Hilbert spaces.
Frequently asked questions
What does it mean to regard a function as a "point"?
Just as a point in the plane can be described by two coordinates and a point in space by three, a function can be regarded as a collection of infinitely many values. A whole function then becomes a single "point," and the totality of functions a single "space." Once we bring in tools for measuring the distance and angle between points (norms and inner products), we can treat the world of functions like geometry. This is the idea from which functional analysis departs.
Why is it convenient to view functions as a space?
Because once functions are laid out as points, the intuitions available for finite-dimensional vectors — closeness, orthogonality, best approximation, expansion in a basis — carry over almost unchanged. A Fourier series becomes "an expansion in orthogonal functions," and least squares becomes "dropping a perpendicular from a point onto a subspace," so topics that looked separate connect in one geometric language. The greatest benefit is being able to translate hard problems into familiar problems about shapes.