2 Basics of Linear Spaces

  A person is solely the concentration of an infinite set of interrelations with another and others, and to separate a person from these relations means to take away any real meaning of the life.

 Vl. Soloviev

A space around us could be described as a three dimensional Euclidean space. To single out a point of that space we need a fixed frame of references and three real numbers, which are coordinates of the point. Similarly to describe a pair of points from our space we could use six coordinates; for three points—nine, end so on. This makes it reasonable to consider Euclidean (linear) spaces of an arbitrary finite dimension, which are studied in the courses of linear algebra.

The basic properties of Euclidean spaces are determined by its linear and metric structures. The linear space (or vector space) structure allows to add and subtract vectors associated to points as well as to multiply vectors by real or complex numbers (scalars).

The metric space structure assign a distance—non-negative real number—to a pair of points or, equivalently, defines a length of a vector defined by that pair. A metric (or, more generally a topology) is essential for definition of the core analytical notions like limit or continuity. The importance of linear and metric (topological) structure in analysis sometime encoded in the formula:

On the other hand we could observe that many sets admit a sort of linear and metric structures which are linked each other. Just few among many other examples are:

• The set of convergent sequences;
• The set of continuous functions on [0,1].

It is a very mathematical way of thinking to declare such sets to be spaces and call their elements points.

But shall we lose all information on a particular element (e.g. a sequence {1/n}) if we represent it by a shapeless and size-less “point” without any inner configuration? Surprisingly not: all properties of an element could be now retrieved not from its inner configuration but from interactions with other elements through linear and metric structures. Such a “sociological” approach to all kind of mathematical objects was codified in the abstract category theory.

Another surprise is that starting from our three dimensional Euclidean space and walking far away by a road of abstraction to infinite dimensional Hilbert spaces we are arriving just to yet another picture of the surrounding space—that time on the language of quantum mechanics.

  The distance from Manchester to Liverpool is 35 miles—just about the mileage in the opposite direction!

  A tourist guide to England

2.1 Banach spaces (basic definitions only)

The following definition generalises the notion of distance known from the everyday life.

The following notion is a useful specialisation of metric adopted to the linear structure.

The connection between norm and metric is as follows:

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  (a)    (b)    

Figure 1: Triangle inequality in metric (a) and normed (b) spaces.

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An important notions known from real analysis are limit and convergence. Particularly we usually wish to have enough limiting points for all “reasonable” sequences.

For example, the set of integers ℤ and reals ℝ with the natural distance functions are complete spaces, but the set of rationals ℚ is not. The complete normed spaces deserve a special name.

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(i)    (ii)   (iii)

Figure 2: Different unit balls defining norms in ℝ2 from Example 9.

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  —We need an extra space to accommodate this product!

  A manager to a shop assistant

2.2 Hilbert spaces

Although metric and norm capture important geometric information about linear spaces they are not sensitive enough to represent such geometric characterisation as angles (particularly orthogonality). To this end we need a further refinements.

From courses of linear algebra known that the scalar product ⟨ x,y ⟩= x₁ y₁ + ⋯ + x_n y_n is important in a space ℝⁿ and defines a norm ||x||²=⟨ x,x ⟩. Here is a suitable generalisation:

Last two properties of the scalar product is oftenly encoded in the phrase: “it is linear in the first variable if we fix the second and anti-linear in the second if we fix the first”.

Now we state, probably, the most important inequality in analysis.

Again complete inner product spaces deserve a special name

The relations between spaces introduced so far are as follows:

How can we tell if a given norm comes from an inner product?

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Figure 3: To the parallelogram identity.

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  Divide and rule!

  Old but still much used recipe

2.3 Subspaces

To study Hilbert spaces we may use the traditional mathematical technique of analysis and synthesis: we split the initial Hilbert spaces into smaller and probably simpler subsets, investigate them separately, and then reconstruct the entire picture from these parts.

As known from the linear algebra, a linear subspace is a subset of a linear space is its subset, which inherits the linear structure, i.e. possibility to add vectors and multiply them by scalars. In this course we need also that subspaces inherit topological structure (coming either from a norm or an inner product) as well.

We also wish that the both inhered structures (linear and topological) should be in agreement, i.e. the subspace should be complete. Such inheritance is linked to the property be closed.

A subspace need not be closed—for example the sequence

and x_n=(1, 1/2,…, 1/n, 0, 0,…)∈ c₀₀ converges to x thus x∈ c₀₀ ⊂ l₂.

Hence c₀₀ is an incomplete inner product space, with inner product ⟨ x,y ⟩=∑₁^∞x_k ȳ_k (this is a finite sum!) as it is not closed in l₂.

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(a)     (b) 

Figure 4: Jump function on (b) as a L2 limit of continuous functions from (a).

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Similarly C[0,1] with inner product norm ||f||=(∫₀¹ | f(t) |² dt)^1/2 is incomplete—take the large space X of functions continuous on [0,1] except for a possible jump at 1/2 (i.e. left and right limits exists but may be unequal and f(1/2)=lim_t→1/2+ f(t). Then the sequence of functions defined on Figure 4(a) has the limit shown on Figure 4(b) since:

Obviously f∈C[0,1]∖C[0,1].

Similarly the space C[a,b] is incomplete for any a<b if equipped by the inner product and the corresponding norm:

It is practical to realise L₂[a,b] as a certain space of “functions” with the inner product defined via an integral. There are several ways to do that and we mention just two:

1. Elements of L₂[a,b] are equivalent classes of Cauchy sequences f⁽ⁿ⁾ of functions from C[a,b].
2. Let integration be extended from the Riemann definition to the wider Lebesgue integration (see Section 13). Let L be a set of square integrable in Lebesgue sense functions on [a,b] with a finite norm (18). Then L₂[a,b] is a quotient space of L with respect to the equivalence relation f∼ g ⇔ ||f−g||₂=0 .
   Example 23  Let the Cantor function on [0,1] be defined as follows:

   f(t)=

   ⎧ ⎨ ⎩

   1, t∈ ℚ;           0, t∈ ℝ∖ℚ.

   This function is not integrable in the Riemann sense but does have the Lebesgue integral. The later however is equal to 0 and as an L₂-function the Cantor function equivalent to the function identically equal to 0.
3. The third possibility is to map L₂(ℝ) onto a space of “true” functions but with an additional structure. For example, in quantum mechanics it is useful to work with the Segal–Bargmann space of analytic functions on ℂ with the inner product [, , ]:

   ⟨ f1,f2  ⟩=

   ∫

   ℂ

   f1(z) f2(z)e

   − ⎪ ⎪ z  ⎪ ⎪ 2

   d z.

  All good things are covered by a thick layer of chocolate (well, if something is not yet–it certainly will)

2.4 Linear spans

As was explained into introduction 2, we describe “internal” properties of a vector through its relations to other vectors. For a detailed description we need sufficiently many external reference points.

Let A be a subset (finite or infinite) of a normed space V. We may wish to upgrade it to a linear subspace in order to make it subject to our theory.

Consequently CLin(A) is the set of all limiting points of finite linear combination of elements of A.

The following simple result will be used later many times without comments.