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Definition and illustration

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The statement of the theorem

One of the most familiar examples of a Hilbert space is the Euclidean vector space consisting of three-dimensional vectors, denoted by R3\mathbf {R} ^{3} , and equipped with the dot product. The dot product takes two vectors x and y, and produces a real number x ⋅ y. If x and y are represented in Cartesian coordinates, then the dot product is defined by: (x1x2x3)⋅(y1y2y3)=x1y1+x2y2+x3y3 .{\begin{pmatrix}x_{1}\\x_{2}\\x_{3}\end{pmatrix}}\cdot {\begin{pmatrix}y_{1}\\y_{2}\\y_{3}\end{pmatrix}}=x_{1}y_{1}+x_{2}y_{2}+x_{3}y_{3}\,. The dot product satisfies the properties: - It is symmetric in x and y: x ⋅ y = y ⋅ x. - It is linear in its first argument: (ax_{1} + bx_{2}) ⋅ y = a(x_{1} ⋅ y) + b(x_{2} ⋅ y) for any scalarsa, b, and vectors x_{1}, x_{2}, and y. - It is positive definite: for all vectors x, x ⋅ x ≥ 0 , with equality if and only ifx = 0. An operation on pairs of vectors that, like the dot product, satisfies these three properties is known as a (real) inner product. A vector space equipped with such an inner product is known as a (real) inner product space. Every finite-dimensional inner product space is also a Hilbert space. The basic feature of the dot product that connects it with Euclidean geometry is that it is related to both the length (or norm) of a vector, denoted ‖x‖, and to the angle θ between two vectors x and y by means of the formula x⋅y=∥x∥∥y∥ cos⁡θ .\mathbf {x} \cdot \mathbf {y} =\left\|\mathbf {x} \right\|\left\|\mathbf {y} \right\|\,\cos \theta \,. Completeness means that a series of vectors (in blue) results in a well-defined net displacement vector (in orange). Multivariable calculus in Euclidean space relies on the ability to compute limits, and to have useful criteria for concluding that limits exist. A mathematical series ∑n=0∞xn\sum _{n=0}^{\infty }\mathbf {x} _{n} consisting of vectors in R^{3} is absolutely convergent provided that the sum of the lengths converges as an ordinary series of real numbers: ∑k=0∞∥xk∥<∞ .\sum _{k=0}^{\infty }\|\mathbf {x} _{k}\|<\infty \,. Just as with a series of scalars, a series of vectors that converges absolutely also converges to some limit vector L in the Euclidean space, in the sense that lim⁡N→∞∥L−∑k=0Nxk∥=0.\lim _{N\to \infty }\left\|\mathbf {L} -\sum _{k=0}^{N}\mathbf {x} _{k}\right\|=0. This property expresses the completeness of Euclidean space: that a series that converges absolutely also converges in the ordinary sense. Hilbert spaces are often taken over the complex numbers. The complex plane denoted by C is equipped with a notion of magnitude, the complex modulus|z|, which is defined as the square root of the product of z with its complex conjugate: ∣z∣2=zz‾ .|z|^{2}=z{\overline {z}}\,. If z = x + iy is a decomposition of z into its real and imaginary parts, then the modulus is the usual Euclidean two-dimensional length: ∣z∣=x2+y2 .|z|={\sqrt {x^{2}+y^{2}}}\,. The inner product of a pair of complex numbers z and w is the product of z with the complex conjugate of w: ⟨z,w⟩=zw‾ .\langle z,w\rangle =z{\overline {w}}\,. This is complex-valued. The real part of ⟨z, w⟩ gives the usual two-dimensional Euclidean dot product. A second example is the space C^{2} whose elements are pairs of complex numbers z = (z_{1}, z_{2}). Then an inner product of z with another such vector w = (w_{1}, w_{2}) is given by ⟨z,w⟩=z1w‾1+z2w‾2 .\langle z,w\rangle =z_{1}{\overline {w}}_{1}+z_{2}{\overline {w}}_{2}\,. The real part of ⟨z, w⟩ is then the four-dimensional Euclidean dot product. This inner product is Hermitian symmetric, which means that the result of interchanging z and w is the complex conjugate: ⟨w,z⟩=⟨z,w⟩‾ .\langle w,z\rangle ={\overline {\langle z,w\rangle }}\,. A Hilbert space is a real or complexinner product space that is also a complete metric space with respect to the distance function induced by the inner product. To say that a complex vector space H is a complex inner product space means that there is an inner product ⟨x,y⟩\langle x,y\rangle associating a complex number to each pair of elements x,yx,y of H that satisfies the following properties: - The inner product is conjugate symmetric; that is, the inner product of a pair of elements is equal to the complex conjugate of the inner product of the swapped elements: ⟨y,x⟩=⟨x,y⟩‾ .\langle y,x\rangle ={\overline {\langle x,y\rangle }}\,. Importantly, this implies that ⟨x,x⟩\langle x,x\rangle is a real number. - The inner product is linear in its first argument. For all complex numbers aa and b,b, ⟨ax1+bx2,y⟩=a⟨x1,y⟩+b⟨x2,y⟩ .\langle ax_{1}+bx_{2},y\rangle =a\langle x_{1},y\rangle +b\langle x_{2},y\rangle \,. - The inner product of an element with itself is positive definite: ⟨x,x⟩>0 if x≠0,⟨x,x⟩=0 if x=0 .{\begin{alignedat}{4}\langle x,x\rangle >0&\quad {\text{ if }}x\neq 0,\\\langle x,x\rangle =0&\quad {\text{ if }}x=0\,.\end{alignedat}} It follows from properties 1 and 2 that a complex inner product is antilinear, also called conjugate linear, in its second argument, meaning that ⟨x,ay1+by2⟩=aˉ⟨x,y1⟩+bˉ⟨x,y2⟩ .\langle x,ay_{1}+by_{2}\rangle ={\bar {a}}\langle x,y_{1}\rangle +{\bar {b}}\langle x,y_{2}\rangle \,. A real inner product space is defined in the same way, except that H is a real vector space and the inner product takes real values. Such an inner product will be a bilinear map and (H,H,⟨⋅,⋅⟩)(H,H,\langle \cdot ,\cdot \rangle ) will form a dual system. Illustration of triangle inequality with distance function on each side The norm is the real-valued function ∥x∥=⟨x,x⟩ ,\|x\|={\sqrt {\langle x,x\rangle }}\,, and the distance dd between two points x,yx,y in H is defined in terms of the norm by d(x,y)=∥x−y∥=⟨x−y,x−y⟩ .d(x,y)=\|x-y\|={\sqrt {\langle x-y,x-y\rangle }}\,. Here, d(x,y)d(x,y) is a distance function meaning firstly that it is symmetric in xx and y,y, secondly that the distance between xx and itself is zero, and otherwise the distance between xx and yy must be positive, and lastly that the triangle inequality holds, meaning that the length of one leg of a triangle xyz cannot exceed the sum of the lengths of the other two legs: d(x,z)≤d(x,y)+d(y,z) .d(x,z)\leq d(x,y)+d(y,z)\,. This last property is ultimately a consequence of the more fundamental Cauchy–Schwarz inequality, which asserts ∣⟨x,y⟩∣≤∥x∥∥y∥\left|\langle x,y\rangle \right|\leq \|x\|\|y\| with equality if and only if xx and yy are linearly dependent. With a distance function defined in this way, any inner product space is a metric space, and sometimes is known as a pre-Hilbert space. Any pre-Hilbert space that is additionally also a complete space is a Hilbert space. The completeness of H is expressed using a form of the Cauchy criterion for sequences in H: a pre-Hilbert space H is complete if every Cauchy sequenceconverges with respect to this norm to an element in the space. Completeness can be characterized by the following equivalent condition: if a series of vectors ∑k=0∞uk\sum _{k=0}^{\infty }u_{k} converges absolutely in the sense that ∑k=0∞∥uk∥<∞ ,\sum _{k=0}^{\infty }\|u_{k}\|<\infty \,, then the series converges in H, in the sense that the partial sums converge to an element of H. As a complete normed space, Hilbert spaces are by definition also Banach spaces. As such they are topological vector spaces, in which topological notions like the openness and closedness of subsets are well defined. Of special importance is the notion of a closed linear subspace of a Hilbert space that, with the inner product induced by restriction, is also complete (being a closed set in a complete metric space) and therefore a Hilbert space in its own right. The sequence space ℓ2{\textstyle \ell ^{2}} consists of all infinite sequencesz = (z_{1}, z_{2}, ...) of complex numbers such that the series of its squared norms converges: ∑n=1∞∣zn∣2\sum _{n=1}^{\infty }|z_{n}|^{2} The inner product on l^{2} is defined by: ⟨z,w⟩=∑n=1∞znw‾n ,\langle \mathbf {z} ,\mathbf {w} \rangle =\sum _{n=1}^{\infty }z_{n}{\overline {w}}_{n}\,, The series for the inner product converges as a consequence of the Cauchy–Schwarz inequality and the assumed convergence of the two series of squared norms. Completeness of the space holds provided that whenever a series of elements from ℓ2{\textstyle \ell ^{2}} converges absolutely (in norm), then it converges to an element of ℓ2{\textstyle \ell ^{2}} . The proof is basic in mathematical analysis, and permits mathematical series of elements of the space to be manipulated with the same ease as series of complex numbers (or vectors in a finite-dimensional Euclidean space). - ^ ^{a}^{b}Axler 2024, p. 182, 6A Inner Products and Norms - ^However, some sources call finite-dimensional spaces with these properties pre-Hilbert spaces, reserving the term "Hilbert space" for infinite-dimensional spaces; see, e.g., Levitan 2001. - ^Olver, Peter J.; Shakiban, Chehrzad (2018), Applied Linear Algebra, Undergraduate Texts in Mathematics, Springer International Publishing, p. 137, doi:10.1007/978-3-319-91041-3, ISBN9783319910413 - ^Marsden 1974, §2.8 - ^Kainth (2023). For the completeness of Euclidean space, see Definition 4.37 and Example 4.38, p. 108; for the equivalence of completeness with the property that absolutely convergent series converge, see Theorem 4.44, p. 110. - ^ ^{a}^{b}Vince, John (2018), Imaginary Mathematics for Computer Science, Springer International Publishing, doi:10.1007/978-3-319-94637-5, ISBN9783319946375 - ^ ^{a}^{b}^{c}^{d}Andrilli, Stephen; Hecker, David (2010), Elementary Linear Algebra, Elsevier, pp. 446–447, ISBN9780080886251 - ^The mathematical material in this section can be found in any good textbook on functional analysis, such as Dieudonné (1960), Hewitt & Stromberg (1965), Reed & Simon (1980) or Rudin (1987). - ^ ^{a}^{b}^{c}^{d}Axler (2024), pp. 183–184. - ^Axler (2024), p. 185, Properties 6(d) and 6(e). - ^Schaefer & Wolff 1999, pp. 122–202 - ^Axler (2024), p. 186. - ^Dieudonné (1960), Section V.1. - ^Dieudonné (1960), Section III.1. - ^ ^{a}^{b}Dieudonné (1960), Section VI.2. - ^Roman 2008, p. 327 - ^Roman 2008, p. 330 Theorem 13.8 - ^Kainth (2023), p. 108, Definition 4.37. - ^Schaefer & Wolff 1999, pp. 40–41. - ^ ^{a}^{b}Stein & Shakarchi 2005, p. 163 - ^Deitmar 2005, p. 26. - ^Dieudonné 1960 Cite error: There are tags or {{efn}} templates on this page, but the references will not show without a {{reflist|group=lower-alpha}} template or {{notelist}} template (see the help page).