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Geometric interpretation of inner product as angle between two vectors

In mathematics, an inner product space is a vector space with the additional structure called an inner product. This additional structure associates each pair of vectors in the space with a scalar quantity known as the inner product of the vectors. Inner products allow the rigorous introduction of intuitive geometrical notions such as the length of a vector or the angle between two vectors. They also provide the means of defining orthogonality between vectors (zero inner product). Inner product spaces generalize Euclidean spaces (in which the inner product is the dot product, also known as the scalar product) to vector spaces of any (possibly infinite) dimension, and are studied in functional analysis.

An inner product space is sometimes also called a pre-Hilbert space, since its completion with respect to the metric, induced by its inner product, is a Hilbert space. That is, if a pre-Hilbert space is complete with respect to the metric arising from its inner product (and norm), then it is called a Hilbert space.

Inner product spaces were referred to as unitary spaces in earlier work, although this terminology is now rarely used.

Contents

Definition

In this article, the field of scalars denoted \mathbb{F} is either the field of real numbers \mathbb{R} or the field of complex numbers \mathbb{C}.

Formally, an inner product space is a vector space V over the field \mathbb{F} together with an inner product, i.e., with a map

 \langle \cdot, \cdot \rangle : V \times V \rightarrow \mathbb{F}

that satisfies the following three axioms for all vectors x,y,z \in V and all scalars a \in \mathbb{F}:

\langle x,y\rangle =\overline{\langle y,x\rangle}.
\langle ax,y\rangle= a \langle x,y\rangle.
\langle x+y,z\rangle= \langle x,z\rangle+ \langle y,z\rangle.
\langle x,x\rangle \geq 0 with equality only for x = 0.

Notice that conjugate symmetry implies that \langle x,x \rangle is real for all x, since we have \langle x,x \rangle = \overline{\langle x,x \rangle}.

Conjugate symmetry and linearity in the first variable gives

 \langle x, a y \rangle = \overline{\langle a y, x \rangle} = \overline{a} \overline{\langle y, x \rangle} = \overline{a} \langle x, y \rangle
 \langle x, y + z \rangle = \overline{\langle y + z, x \rangle} = \overline{\langle y, x \rangle} + \overline{\langle z, x \rangle} = \langle x, y \rangle + \langle x, z \rangle,

so an inner product is a sesquilinear form. Conjugate symmetry is also called Hermitian symmetry, and a conjugate symmetric sesquilinear form is called a Hermitian form While the above axioms are more mathematically economical, a compact verbal definition of an inner product is a positive-definite Hermitian form.

In the case of  \mathbb{F} = \mathbb{R} , conjugate-symmetric reduces to symmetric, and sesquilinear reduces to bilinear. So, an inner product on a real vector space is a positive-definite symmetric bilinear form.

From the linearity property it is derived that x = 0 implies \langle x,x \rangle = 0, while from the positive-definiteness axiom we obtain the converse, \langle x,x \rangle = 0 implies x = 0. Combining these two, we have the property that \langle x,x \rangle = 0 if and only if x = 0.

The property of an inner product space V that

 \langle x+y,z\rangle= \langle x,z\rangle+ \langle y,z\rangle and  \langle x,y+z\rangle = \langle x,y\rangle + \langle x,z\rangle

is known as additivity.

Remark: Some authors, especially in physics and matrix algebra, prefer to define the inner product and the sesquilinear form with linearity in the second argument rather than the first. Then the first argument becomes conjugate linear, rather than the second. In those disciplines we would write the product \langle x,y\rangle as \langle y|x\rangle (the bra-ket notation of quantum mechanics), respectively yTx (dot product as a case of the convention of forming the matrix product AB as the dot products of rows of A with columns of B). Here the kets and columns are identified with the vectors of V and the bras and rows with the dual vectors or linear functionals of the dual space V*, with conjugacy associated with duality. This reverse order is now occasionally followed in the more abstract literature, e.g., Emch [1972], taking \langle x,y\rangle to be conjugate linear in x rather than y. A few instead find a middle ground by recognizing both \langle , \rangle and \langle | \rangle as distinct notations differing only in which argument is conjugate linear.

There are various technical reasons why it is necessary to restrict the basefield to \mathbb{R} and \mathbb{C} in the definition. Briefly, the basefield has to contain an ordered subfield (in order for non-negativity to make sense) and therefore has to have characteristic equal to 0. This immediately excludes finite fields. The basefield has to have additional structure, such as a distinguished automorphism. More generally any quadratically closed subfield of \mathbb{R} or \mathbb{C} will suffice for this purpose, e.g., the algebraic numbers, but when it is a proper subfield (i.e., neither \mathbb{R} nor \mathbb{C}) even finite-dimensional inner product spaces will fail to be metrically complete. In contrast all finite-dimensional inner product spaces over \mathbb{R} or \mathbb{C}, such as those used in quantum computation, are automatically metrically complete and hence Hilbert spaces.

In some cases we need to consider non-negative semi-definite sesquilinear forms. This means that \langle x,x\rangle is only required to be non-negative. We show how to treat these below.

Examples

  • A trivial example is the real numbers with the standard multiplication as the inner product
\langle x,y\rangle := xy.
More generally any Euclidean space \mathbb{R} n with the dot product is an inner product space
\langle (x_1,\ldots, x_n),(y_1,\ldots, y_n)\rangle := \sum_{i=1}^{n} x_i y_i = x_1 y_1 + \cdots + x_n y_n.
  • The general form of an inner product on \mathbb{C} n is given by:
\langle \mathbf{x},\mathbf{y}\rangle := \mathbf{y}^*\mathbf{M}\mathbf{x}
with M any symmetric positive-definite matrix, and y* the conjugate transpose of y. For the real case this corresponds to the dot product of the results of directionally differential scaling of the two vectors, with positive scale factors and orthogonal directions of scaling. Up to an orthogonal transformation it is a weighted-sum version of the dot product, with positive weights.
  • The article on Hilbert space has several examples of inner product spaces wherein the metric induced by the inner product yields a complete metric space. An example of an inner product which induces an incomplete metric occurs with the space C[ab] of continuous complex valued functions on the interval [ab]. The inner product is
 \langle f , g \rangle := \int_a^b f(t) \overline{g(t)} \, dt.
This space is not complete; consider for example, for the interval [−1,1] the sequence of "step" functions { fk }k where
  • fk(t) is 0 for t in the subinterval [−1,0]
  • fk(t) is 1 for t in the subinterval [1/k, 1]
  • fk is affine in [0, 1/k].
This sequence is a Cauchy sequence which does not converge to a continuous function.
 \langle X, Y \rangle := \operatorname{E}(X Y)
is an inner product. In this case, <X, X>=0 if and only if Pr(X=0)=1 (i.e., X=0 almost surely). This definition of expectation as inner product can be extended to random vectors as well.

Norms on inner product spaces

Inner product spaces have a naturally defined norm

 \|x\| =\sqrt{\langle x, x\rangle}.

This is well defined by the nonnegativity axiom of the definition of inner product space. The norm is thought of as the length of the vector x. Directly from the axioms, we can prove the following:

 |\langle x,y\rangle| \leq \|x\| \cdot \|y\|
with equality if and only if x and y are linearly dependent. This is one of the most important inequalities in mathematics. It is also known in the Russian mathematical literature as the Cauchy–Bunyakowski–Schwarz inequality.
Because of its importance, its short proof should be noted.
It is trivial to prove the inequality true in the case y = 0. Thus we assume ⟨y, y⟩ is nonzero, giving us the following:
 \lambda = \langle y , y \rangle^{-1} \langle x, y\rangle
 0 \leq \langle x -\lambda y, x -\lambda y \rangle = \langle x, x\rangle - \langle y , y \rangle^{-1} | \langle x,y\rangle|^2.
The complete proof can be obtained by multiplying out this result.
  • Orthogonality: The geometric interpretation of the inner product in terms of angle and length, motivates much of the geometric terminology we use in regard to these spaces. Indeed, an immediate consequence of the Cauchy-Schwarz inequality is that it justifies defining the angle between two non-zero vectors x and y (at least in the case F = \mathbb{R}) by the identity
\operatorname{angle}(x,y) = \arccos \frac{\langle x, y \rangle}{\|x\| \cdot \|y\|}.
We assume the value of the angle is chosen to be in the interval [0, +π]. This is in analogy to the situation in two-dimensional Euclidean space. Correspondingly, we will say that non-zero vectors x, y of V are orthogonal if and only if their inner product is zero.
 \|r \cdot x\| = |r| \cdot \| x\|.
The homogeneity property is completely trivial to prove.
 \|x + y\| \leq \|x \| + \|y\|.
The last two properties show the function defined is indeed a norm.
Because of the triangle inequality and because of axiom 2, we see that ||·|| is a norm which turns V into a normed vector space and hence also into a metric space. The most important inner product spaces are the ones which are complete with respect to this metric; they are called Hilbert spaces. Every inner product V space is a dense subspace of some Hilbert space. This Hilbert space is essentially uniquely determined by V and is constructed by completing V.
 \|x + y\|^2 + \|x - y\|^2 = 2\|x\|^2 + 2\|y\|^2.
 \|x\|^2 + \|y\|^2 = \|x+y\|^2.
The proofs of both of these identities require only expressing the definition of norm in terms of the inner product and multiplying out, using the property of additivity of each component. Alternatively, both can be seen as consequences of the identity
 \|x + y\|^2 = \|x\|^2 + \|y\|^2 + 2 \real \langle x , y \rangle.
which is a form of the law of cosines, and is proved as before.
The name Pythagorean theorem arises from the geometric interpretation of this result as an analogue of the theorem in synthetic geometry. Note that the proof of the Pythagorean theorem in synthetic geometry is considerably more elaborate because of the paucity of underlying structure. In this sense, the synthetic Pythagorean theorem, if correctly demonstrated is deeper than the version given above.
An induction on the Pythagorean theorem yields:
  • If x1, ..., xn are orthogonal vectors, that is, \langle x_j,x_k\rangle=0 for distinct indices j, k, then
 \sum_{i=1}^n \|x_i\|^2 = \left\|\sum_{i=1}^n x_i \right\|^2.
In view of the Cauchy-Schwarz inequality, we also note that \langle\cdot,\cdot\rangle is continuous from V × V to F. This allows us to extend Pythagoras' theorem to infinitely many summands:
  • Parseval's identity: Suppose V is a complete inner product space. If {xk} are mutually orthogonal vectors in V then
 \sum_{i=1}^\infty\|x_i\|^2 = \left\|\sum_{i=1}^\infty x_i\right\|^2,
provided the infinite series on the left is convergent. Completeness of the space is needed to ensure that the sequence of partial sums
 S_k = \sum_{i=1}^k x_i,
which is easily shown to be a Cauchy sequence, is convergent.

Orthonormal sequences

Let V be a finite dimensional inner product space of dimension n. Recall that every basis of V consists of exactly n linearly independent vectors. Using the Gram-Schmidt Process we may start with an arbitrary basis and transform it into an orthonormal basis. That is, into a basis in which all the elements are orthogonal and have unit norm. In symbols, a basis \textstyle {\{e_1,\ldots,e_n\}} is orthonormal if \textstyle \langle e_i, e_j\rangle=0 if \textstyle i\neq j and \textstyle \langle e_i, e_i\rangle = ||e_i|| = 1 for each i.

This definition of orthonormal basis generalizes to the case of infinite dimensional inner product spaces in the following way. Let V be a any inner product space. Then an collection \textstyle E=\{e_{\alpha}\}_{\alpha \in A} is a basis for V if the subspace of V generated by finite linear combinations of elements of E is dense in V (in the norm induced by the inner product). We say that E is an orthonormal basis for V if it is a basis and \textstyle \langle e_{\alpha}, e_{\beta}\rangle=0 if \textstyle \alpha \neq \beta and \textstyle \langle e_{\alpha}, e_{\alpha}\rangle = ||e_{\alpha}|| = 1 for all \textstyle \alpha,\beta \in A.

Using an infinite-dimensional analog of the Gram-Schmidt process one may show:

Theorem. Any separable inner product space V has an orthonormal basis.

Using the Hausdorff Maximal Principle and the fact that in a complete inner product space orthogonal projection onto linear subspaces is well-defined, one may also show that

Theorem. Any complete inner product space V has an orthonormal basis.

The two previous theorems raise the question of whether all inner product spaces have an orthonormal basis. The answer, it turns out is negative. This is a non-trivial result, and is proved below. The following proof is taken from Halmos's A Hilbert Space Problem Book (see the references).

Parseval's identity leads immediately to the following theorem:

Theorem. Let V be a separable inner product space and {ek}k an orthonormal basis of V. Then the map

 x \mapsto \{\langle e_k, x\rangle\}_{k \in \mathbb{N}}

is an isometric linear map V 2 with a dense image.

This theorem can be regarded as an abstract form of Fourier series, in which an arbitrary orthonormal basis plays the role of the sequence of trigonometric polynomials. Note that the underlying index set can be taken to be any countable set (and in fact any set whatsoever, provided  2 is defined appropriately, as is explained in the article Hilbert space). In particular, we obtain the following result in the theory of Fourier series:

Theorem. Let V be the inner product space C[ − π,π]. Then the sequence (indexed on set of all integers) of continuous functions

ek(t) = (2π) − 1 / 2eikt

is an orthonormal basis of the space C[ − π,π] with the L2 inner product. The mapping

 f \mapsto \frac{1}{\sqrt{2 \pi}} \left\{\int_{-\pi}^\pi f(t) e^{-i k t} \, dt \right\}_{k \in \mathbb{Z}}

is an isometric linear map with dense image.

Orthogonality of the sequence {ek}k follows immediately from the fact that if k ≠ j, then

 \int_{-\pi}^\pi e^{-i (j-k) t} \, dt = 0.

Normality of the sequence is by design, that is, the coefficients are so chosen so that the norm comes out to 1. Finally the fact that the sequence has a dense algebraic span, in the inner product norm, follows from the fact that the sequence has a dense algebraic span, this time in the space of continuous periodic functions on [ − π,π] with the uniform norm. This is the content of the Weierstrass theorem on the uniform density of trigonometric polynomials.

Operators on inner product spaces

Several types of linear maps A from an inner product space V to an inner product space W are of relevance:

  • Continuous linear maps, i.e., A is linear and continuous with respect to the metric defined above, or equivalently, A is linear and the set of non-negative reals {||Ax||}, where x ranges over the closed unit ball of V, is bounded.
  • Symmetric linear operators, i.e., A is linear and ⟨Ax, y⟩ = ⟨x, A y⟩ for all x, y in V.
  • Isometries, i.e., A is linear and ⟨Ax, Ay⟩ = ⟨x, y⟩ for all x, y in V, or equivalently, A is linear and ||Ax|| = ||x|| for all x in V. All isometries are injective. Isometries are morphisms between inner product spaces, and morphisms of real inner product spaces are orthogonal transformations (compare with orthogonal matrix).
  • Isometrical isomorphisms, i.e., A is an isometry which is surjective (and hence bijective). Isometrical isomorphisms are also known as unitary operators (compare with unitary matrix).

From the point of view of inner product space theory, there is no need to distinguish between two spaces which are isometrically isomorphic. The spectral theorem provides a canonical form for symmetric, unitary and more generally normal operators on finite dimensional inner product spaces. A generalization of the spectral theorem holds for continuous normal operators in Hilbert spaces.

Degenerate inner products

If V is a vector space and ⟨ , ⟩ a semi-definite sesquilinear form, then the function ||x|| = ⟨xx1/2 makes sense and satisfies all the properties of norm except that ||x|| = 0 does not imply x = 0 (such a functional is then called a semi-norm). We can produce an inner product space by considering the quotient W = V/{ x : ||x|| = 0}. The sesquilinear form ⟨ , ⟩ factors through W.

This construction is used in numerous contexts. The Gelfand–Naimark–Segal construction is a particularly important example of the use of this technique. Another example is the representation of semi-definite kernels on arbitrary sets.

See also

References

  • Axler, Sheldon (1997), Linear Algebra Done Right (2nd ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-387-98258-8  
  • Emch, Gerard G. (1972), Algebraic methods in statistical mechanics and quantum field theory, Wiley-Interscience, ISBN 978-0-471-23900-0  
  • Young, Nicholas (1988), An introduction to Hilbert space, Cambridge University Press, ISBN 978-0-521-33717-5  
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