A Hilbert space<\/strong> is a complete vector space equipped with an inner product that allows the measurement of lengths, angles, and orthogonality. It generalizes the familiar Euclidean space to finite- and infinite-dimensional settings and forms one of the central mathematical structures in functional analysis, quantum mechanics, signal processing, and machine learning.<\/p>\n\n\n\n A Hilbert space<\/strong> $ H $ is a vector space over the field of real numbers $\\mathbb{R}$ or complex numbers $\\mathbb{C}$ together with an inner product<\/strong><\/p>\n\n\n\n $$ where $\\mathbb{F} = \\mathbb{R}$ or $\\mathbb{C},$ satisfying the following axioms:<\/p>\n\n\n\n The inner product induces a norm<\/strong><\/p>\n\n\n\n \\sqrt{\\langle x,x\\rangle}, which defines a metric<\/p>\n\n\n\n [ A Hilbert space is complete<\/strong>, meaning that every Cauchy sequence converges to an element of the space.<\/p>\n\n\n\n Completeness is the defining feature distinguishing Hilbert spaces from general inner-product spaces.<\/p>\n\n\n\n Every Cauchy sequence<\/p>\n\n\n\n [ has a limit<\/p>\n\n\n\n [ This guarantees that limits of convergent processes remain inside the space.<\/p>\n\n\n\n The inner product generalizes the Euclidean dot product.<\/p>\n\n\n\n It allows one to define:<\/p>\n\n\n\n For vectors (x,y),<\/p>\n\n\n\n \\frac{\\operatorname{Re}\\langle x,y\\rangle} The induced norm satisfies<\/p>\n\n\n\n Thus,<\/p>\n\n\n\n [ Two vectors are orthogonal if<\/p>\n\n\n\n [ Orthogonality extends the familiar notion of perpendicular vectors.<\/p>\n\n\n\n A Hilbert space possesses an orthonormal basis<\/p>\n\n\n\n [ Every vector can be uniquely represented as<\/p>\n\n\n\n \\sum_i In infinite-dimensional spaces this sum converges in norm.<\/p>\n\n\n\n For an orthonormal basis,<\/p>\n\n\n\n \\sum_i This generalizes the Pythagorean theorem.<\/p>\n\n\n\n If (M) is a closed subspace of a Hilbert space, every vector (x) can be uniquely decomposed as<\/p>\n\n\n\n m+n, where<\/p>\n\n\n\n Thus,<\/p>\n\n\n\n P_Mx, where (P_M) denotes the orthogonal projection.<\/p>\n\n\n\n Every continuous linear functional<\/p>\n\n\n\n [ can be represented uniquely as<\/p>\n\n\n\n \\langle x,y\\rangle for some unique vector<\/p>\n\n\n\n [ This theorem is one of the cornerstones of functional analysis.<\/p>\n\n\n\n For all vectors,<\/p>\n\n\n\n [ Equality holds only when the vectors are linearly dependent.<\/p>\n\n\n\n If<\/p>\n\n\n\n [ then<\/p>\n\n\n\n |x|^2+|y|^2. Unlike Euclidean spaces, Hilbert spaces may have infinitely many independent directions.<\/p>\n\n\n\n Examples include<\/p>\n\n\n\n Hilbert spaces provide the mathematical framework for many areas of modern science:<\/p>\n\n\n\n A Hilbert space is a complete inner-product vector space<\/strong> that extends Euclidean geometry to infinite dimensions. Its key features include:<\/p>\n\n\n\n These properties make Hilbert spaces one of the most powerful and versatile structures in modern mathematics.<\/p>\n","protected":false},"excerpt":{"rendered":" Hilbert Space: Definition and Fundamental Properties A Hilbert space is a complete vector space equipped with an inner product that allows the measurement of lengths, angles, and orthogonality. It generalizes the familiar Euclidean space to finite- and infinite-dimensional settings and forms one of the central mathematical structures in functional analysis, quantum mechanics, signal processing, and…<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-7","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/krdroudnur.cfolks.pl\/index.php\/wp-json\/wp\/v2\/pages\/7","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/krdroudnur.cfolks.pl\/index.php\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/krdroudnur.cfolks.pl\/index.php\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/krdroudnur.cfolks.pl\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/krdroudnur.cfolks.pl\/index.php\/wp-json\/wp\/v2\/comments?post=7"}],"version-history":[{"count":22,"href":"https:\/\/krdroudnur.cfolks.pl\/index.php\/wp-json\/wp\/v2\/pages\/7\/revisions"}],"predecessor-version":[{"id":63,"href":"https:\/\/krdroudnur.cfolks.pl\/index.php\/wp-json\/wp\/v2\/pages\/7\/revisions\/63"}],"wp:attachment":[{"href":"https:\/\/krdroudnur.cfolks.pl\/index.php\/wp-json\/wp\/v2\/media?parent=7"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}Definition<\/h2>\n\n\n\n
\\langle x, y \\rangle : H \\times H \\rightarrow \\mathbb{F},
$$
<\/p>\n\n\n\n\n
[
\\langle ax + by, z \\rangle
= a\\langle x,z\\rangle + b\\langle y,z\\rangle.
]<\/li>\n\n\n\n
[
\\langle x,y\\rangle\\overline{\\langle y,x\\rangle}.
]<\/li>\n\n\n\n
[
\\langle x,x\\rangle \\ge 0,
]
with equality only when
[
x=0.
]<\/li>\n<\/ol>\n\n\n\n[
|x|<\/h1>\n\n\n\n
]<\/p>\n\n\n\n
d(x,y)=|x-y|.
]<\/p>\n\n\n\n
\n\n\n\nFundamental Properties<\/h1>\n\n\n\n
1. Completeness<\/h2>\n\n\n\n
x_1,x_2,\\ldots
]<\/p>\n\n\n\n
x\\in H.
]<\/p>\n\n\n\n
\n\n\n\n2. Inner Product<\/h2>\n\n\n\n
\n
[
\\cos\\theta<\/h1>\n\n\n\n
{|x||y|}.
]<\/p>\n\n\n\n
\n\n\n\n3. Norm<\/h2>\n\n\n\n
\n
|x+y|
\\le
|x|+|y|.
]<\/p>\n\n\n\n
\n\n\n\n4. Orthogonality<\/h2>\n\n\n\n
\\langle x,y\\rangle=0.
]<\/p>\n\n\n\n
\n\n\n\n5. Orthonormal Bases<\/h2>\n\n\n\n
{e_i}.
]<\/p>\n\n\n\n[
x<\/h1>\n\n\n\n
\\langle x,e_i\\rangle e_i.
]<\/p>\n\n\n\n
\n\n\n\n6. Parseval’s Identity<\/h2>\n\n\n\n
[
|x|^2<\/h1>\n\n\n\n
|\\langle x,e_i\\rangle|^2.
]<\/p>\n\n\n\n
\n\n\n\n7. Projection Theorem<\/h2>\n\n\n\n
[
x<\/h1>\n\n\n\n
]<\/p>\n\n\n\n\n
[
m<\/h1>\n\n\n\n
]<\/p>\n\n\n\n
\n\n\n\n8. Riesz Representation Theorem<\/h2>\n\n\n\n
f:H\\rightarrow\\mathbb{F}
]<\/p>\n\n\n\n[
f(x)<\/h1>\n\n\n\n
]<\/p>\n\n\n\n
y\\in H.
]<\/p>\n\n\n\n
\n\n\n\n9. Cauchy\u2013Schwarz Inequality<\/h2>\n\n\n\n
|\\langle x,y\\rangle|
\\le
|x|,|y|.
]<\/p>\n\n\n\n
\n\n\n\n10. Pythagorean Theorem<\/h2>\n\n\n\n
x\\perp y,
]<\/p>\n\n\n\n[
|x+y|^2<\/h1>\n\n\n\n
]<\/p>\n\n\n\n
\n\n\n\n11. Infinite-Dimensional Structure<\/h2>\n\n\n\n
\n
\n\n\n\nTypical Examples<\/h1>\n\n\n\n
Hilbert Space<\/th> Elements<\/th> Inner Product<\/th><\/tr><\/thead> (\\mathbb{R}^n)<\/td> vectors<\/td> (x\\cdot y)<\/td><\/tr> (\\mathbb{C}^n)<\/td> complex vectors<\/td> (\\sum x_i\\overline{y_i})<\/td><\/tr> (\\ell^2)<\/td> square-summable sequences<\/td> (\\sum x_i\\overline{y_i})<\/td><\/tr> (L^2(a,b))<\/td> square-integrable functions<\/td> (\\int_a^b f(x)\\overline{g(x)},dx)<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n
\n\n\n\nImportance in Mathematics and Physics<\/h1>\n\n\n\n
\n
\n\n\n\nSummary<\/h1>\n\n\n\n
\n