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href="/nlab/show/higher+algebra">higher algebra</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homological+algebra">homological algebra</a>, <a class="existingWikiWord" href="/nlab/show/stable+homotopy+theory">stable homotopy theory</a></p> </li> </ul> <h2 id="basic_concepts">Basic concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/ring">ring</a>, <a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E+ring">A-∞ ring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/commutative+ring">commutative ring</a>, <a class="existingWikiWord" href="/nlab/show/E-%E2%88%9E+ring">E-∞ ring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/module">module</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-module">∞-module</a>, <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29-module">(∞,n)-module</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/field">field</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-field">∞-field</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/vector+space">vector space</a>, <a class="existingWikiWord" href="/nlab/show/2-vector+space">2-vector space</a></p> <p><a class="existingWikiWord" href="/nlab/show/rational+vector+space">rational vector space</a></p> <p><a class="existingWikiWord" href="/nlab/show/real+vector+space">real vector space</a></p> <p><a class="existingWikiWord" href="/nlab/show/complex+vector+space">complex vector space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+vector+space">topological vector space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/linear+basis">linear basis</a>,</p> <p><a class="existingWikiWord" href="/nlab/show/orthogonal+basis">orthogonal basis</a>, <a class="existingWikiWord" href="/nlab/show/orthonormal+basis">orthonormal basis</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/linear+map">linear map</a>, <a class="existingWikiWord" href="/nlab/show/antilinear+map">antilinear map</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/matrix">matrix</a> (<a class="existingWikiWord" href="/nlab/show/square+matrix">square</a>, <a class="existingWikiWord" href="/nlab/show/invertible+matrix">invertible</a>, <a class="existingWikiWord" href="/nlab/show/diagonal+matrix">diagonal</a>, <a class="existingWikiWord" href="/nlab/show/hermitian+matrix">hermitian</a>, <a class="existingWikiWord" href="/nlab/show/symmetric+matrix">symmetric</a>, …)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/general+linear+group">general linear group</a>, <a class="existingWikiWord" href="/nlab/show/matrix+group">matrix group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/eigenspace">eigenspace</a>, <a class="existingWikiWord" href="/nlab/show/eigenvalue">eigenvalue</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/inner+product">inner product</a>, <a class="existingWikiWord" href="/nlab/show/Hermitian+form">Hermitian form</a></p> <p><a class="existingWikiWord" href="/nlab/show/Gram-Schmidt+process">Gram-Schmidt process</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hilbert+space">Hilbert space</a></p> </li> </ul> <h2 id="theorems">Theorems</h2> <p>(…)</p> </div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#definition'>Definition</a></li> <li><a href='#background_definitions'>Background definitions</a></li> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="definition">Definition</h2> <p> <div class='num_defn' id='BasisOfAVectorSpace'> <h6>Definition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝕂</mi></mrow><annotation encoding="application/x-tex">\mathbb{K}</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/field">field</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝕂</mi></mrow><annotation encoding="application/x-tex">\mathbb{K}</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/vector+space">vector space</a>, hence a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝕂</mi></mrow><annotation encoding="application/x-tex">\mathbb{K}</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/module">module</a>, a <em><a class="existingWikiWord" href="/nlab/show/linear+basis">linear basis</a></em> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> is</p> <ol> <li> <p>a <a class="existingWikiWord" href="/nlab/show/set">set</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> (of <em>basis elements</em>)</p> </li> <li> <p>a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝕂</mi></mrow><annotation encoding="application/x-tex">\mathbb{K}</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/linear+map">linear</a> <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><munder><mo>⊕</mo><mi>B</mi></munder><mi>𝕂</mi><mover><mo>→</mo><mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace></mrow></mover><mi>V</mi></mrow><annotation encoding="application/x-tex"> \underset{B}{\oplus} \mathbb{K} \xrightarrow{\;\; \simeq \;\;} V </annotation></semantics></math></div></li> </ol> <p>to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> from the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math>-indexed <a class="existingWikiWord" href="/nlab/show/direct+sum">direct sum</a> (the <a class="existingWikiWord" href="/nlab/show/coproduct">coproduct</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝕂</mi></mrow><annotation encoding="application/x-tex">\mathbb{K}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/Vect">Vect</a>) of copies of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝕂</mi></mrow><annotation encoding="application/x-tex">\mathbb{K}</annotation></semantics></math> (canonically regarded as a vector space over itself), also known as the free <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝕂</mi></mrow><annotation encoding="application/x-tex">\mathbb{K}</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/linear+span">linear span</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math>, hence the vector space of free <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝕂</mi></mrow><annotation encoding="application/x-tex">\mathbb{K}</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/linear+combinations">linear combinations</a> of elements of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math>.</p> </div> </p> <p>Hence if a basis for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> exists it means in particular that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/free+module">free module</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝕂</mi></mrow><annotation encoding="application/x-tex">\mathbb{K}</annotation></semantics></math>.</p> <p> <div class='num_remark'> <h6>Remark</h6> <p><strong>(finiteness of linear combinations)</strong> <br /> Beware that a certain finiteness-condition is hidden in Def. <a class="maruku-ref" href="#BasisOfAVectorSpace"></a>: Since a <a class="existingWikiWord" href="/nlab/show/linear+combination">linear combination</a> is defined to be a <a class="existingWikiWord" href="/nlab/show/sum">sum</a> of <a class="existingWikiWord" href="/nlab/show/finite+set">finitely</a> many vectors, a basis of a vector space must be such that every vector in the space is the (unique) combination of <em>finitely</em> many basis elements – even if there are infinitely many elements in the basis.</p> <p>More abstractly this is to do with the appearance of the <a class="existingWikiWord" href="/nlab/show/coproduct">coproduct</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>⨿</mo> <mi>b</mi></msub><mo>=</mo><msub><mo>⊕</mo> <mi>B</mi></msub></mrow><annotation encoding="application/x-tex">\amalg_b = \oplus_B</annotation></semantics></math> (<a class="existingWikiWord" href="/nlab/show/direct+sum">direct sum</a>) in Def. <a class="maruku-ref" href="#BasisOfAVectorSpace"></a> instead of the <a class="existingWikiWord" href="/nlab/show/product">product</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mi>B</mi></msub></mrow><annotation encoding="application/x-tex">\prod_B</annotation></semantics></math>. Many vector spaces in practice arise as (subspaces) of products <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mi>W</mi></msub><mi>𝕂</mi></mrow><annotation encoding="application/x-tex">\prod_W \mathbb{K}</annotation></semantics></math> (<a class="existingWikiWord" href="/nlab/show/function+spaces">function spaces</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi><mo>→</mo><mi>𝕂</mi></mrow><annotation encoding="application/x-tex">W \to \mathbb{K}</annotation></semantics></math>), but if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi></mrow><annotation encoding="application/x-tex">W</annotation></semantics></math> here is not a <a class="existingWikiWord" href="/nlab/show/finite+set">finite set</a> then it is not going to be a basis set.</p> <p>(On the other hand, if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi></mrow><annotation encoding="application/x-tex">W</annotation></semantics></math> <em>is</em> a <a class="existingWikiWord" href="/nlab/show/finite+set">finite set</a>, then we have a <em><a class="existingWikiWord" href="/nlab/show/biproduct">biproduct</a></em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mi>W</mi></msub><mspace width="thinmathspace"></mspace><mo>≃</mo><mspace width="thinmathspace"></mspace><msub><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo> <mi>W</mi></msub><mspace width="thinmathspace"></mspace><mo>≃</mo><mspace width="thinmathspace"></mspace><msub><mo>⊕</mo> <mi>W</mi></msub></mrow><annotation encoding="application/x-tex">\prod_W \,\simeq\, \coprod_W \,\simeq\, \oplus_W</annotation></semantics></math> in the <a class="existingWikiWord" href="/nlab/show/additive+category">additive category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝕂</mi></mrow><annotation encoding="application/x-tex">\mathbb{K}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/Vect">Vect</a>, see <a href="additive+category#ProductsAreBiproducts">there</a>.)</p> </div> </p> <p>Related to this are the following phenomena:</p> <p> <div class='num_remark'> <h6>Remark</h6> <p><strong>(basis and dimension of finitely-generated spaces)</strong> <br /> For every <em>finitely generated</em> vector space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> (Def. <a class="maruku-ref" href="#GeneratedVectorSpace"></a>) it is straightforward to <em><a class="existingWikiWord" href="/nlab/show/constructive+mathematics">construct</a></em> a linear basis, and to see that the <a class="existingWikiWord" href="/nlab/show/cardinality">cardinality</a> of all bases is the same finite <a class="existingWikiWord" href="/nlab/show/natural+number">natural number</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>dim</mi><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">dim(V) \in \mathbb{N}</annotation></semantics></math>, called the <em><a class="existingWikiWord" href="/nlab/show/dimension+of+a+vector+space">dimension</a></em> of the vector space (whence a <em><a class="existingWikiWord" href="/nlab/show/finite-dimensional+vector+space">finite-dimensional vector space</a></em>).</p> </div> </p> <p>On the other hand:</p> <p> <div class='num_remark'> <h6>Remark</h6> <p><strong>(Hamel-bases of infinite-dimensional vector spaces)</strong> <br /> While the definition <a class="maruku-ref" href="#BasisOfAVectorSpace"></a> applies also to not-necessarily finitely generated vector spaces – such as for instance the space of (<a class="existingWikiWord" href="/nlab/show/continuous+function">continuous</a>) <a class="existingWikiWord" href="/nlab/show/functions">functions</a> from a non-finite (<a class="existingWikiWord" href="/nlab/show/topological+space">topological</a>) <a class="existingWikiWord" href="/nlab/show/space">space</a> to the (<a class="existingWikiWord" href="/nlab/show/topological+field">topological</a>) <a class="existingWikiWord" href="/nlab/show/ground+field">ground field</a> – it turns out to be subtle and somewhat ill-behaved in this generality.</p> <p>In fact, in practice infinite-dimensional vector spaces tend to appear and to be understood with <a class="existingWikiWord" href="/nlab/show/extra+structure">extra structure</a> (typically that of <a class="existingWikiWord" href="/nlab/show/topological+vector+spaces">topological vector spaces</a> such as <a class="existingWikiWord" href="/nlab/show/Banach+spaces">Banach spaces</a> or <a class="existingWikiWord" href="/nlab/show/Hilbert+spaces">Hilbert spaces</a>) in which cases there are more appropriate notions of linear bases for them (such as that of <a class="existingWikiWord" href="/nlab/show/Schauder+bases">Schauder bases</a>, which allow infinite-<a class="existingWikiWord" href="/nlab/show/linear+combinations">linear combinations</a> subject to a condition of <a class="existingWikiWord" href="/nlab/show/convergence+of+a+sequence">convergence of a sequence</a>).</p> <p>In order to distinguish the plain notion of basis (Def. <a class="maruku-ref" href="#BasisOfAVectorSpace"></a>) from these more refined notions, one also speaks of <em>Hamel bases</em> here.</p> <p>(This is in honor of <a href="#Hamel1905">Hamel 1905 pp. 460</a> who considered this notion for the special case of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo>=</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">V = \mathbb{R}</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/real+numbers">real numbers</a> regarded as a <a class="existingWikiWord" href="/nlab/show/rational+vector+space">rational vector space</a>, hence over the <a class="existingWikiWord" href="/nlab/show/ground+field">ground field</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝕂</mi><mo>=</mo><mi>ℚ</mi></mrow><annotation encoding="application/x-tex">\mathbb{K} = \mathbb{Q}</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/rational+numbers">rational numbers</a>).</p> <p>Hence, in principle, also a linear basis of a finitely generated vector space is thus a <em>Hamel basis</em>, but rarely called this way unless in the context of infinite-dimensional vector spaces.</p> </div> </p> <p> <div class='num_remark'> <h6>Remark</h6> <p><strong>(basis theorem and dimension)</strong> <br /> For an infinitely-generated vector space it is <em>not</em> in general possible to <em><a class="existingWikiWord" href="/nlab/show/constructive+mathematics">construct</a></em> a (Hamel-)basis, but the <em>existence</em> of such a basis is nevertheless implied, in <a class="existingWikiWord" href="/nlab/show/classical+mathematics">classical mathematics</a>, by <a class="existingWikiWord" href="/nlab/show/Zorn%27s+lemma">Zorn's lemma</a> (essentially a form of the <a class="existingWikiWord" href="/nlab/show/axiom+of+choice">axiom of choice</a>): This is the content of the <em><a class="existingWikiWord" href="/nlab/show/basis+theorem">basis theorem</a></em>.</p> <p>With this classical context understood, it follows that every vector vector space admits a linear basis (even if non-constructible in general) and that each basis is of the same <a class="existingWikiWord" href="/nlab/show/cardinality">cardinality</a>, then called the <a class="existingWikiWord" href="/nlab/show/dimension+of+a+vector+space">dimension</a> of the vector space.</p> </div> </p> <h2 id="background_definitions">Background definitions</h2> <p> <div class='num_defn' id='GeneratedVectorSpace'> <h6>Definition</h6> <p><strong>(generated vector space)</strong> <br /> For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> a vector space, a <a class="existingWikiWord" href="/nlab/show/subset">subset</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo>⊂</mo><mi>V</mi></mrow><annotation encoding="application/x-tex">G \subset V</annotation></semantics></math> of its <a class="existingWikiWord" href="/nlab/show/underlying">underlying</a> set is called a <em>generating set</em> or <em>spanning set</em> if every element of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> can be expressed as a <a class="existingWikiWord" href="/nlab/show/linear+combination">linear combination</a> of elements of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>, hence if the <a class="existingWikiWord" href="/nlab/show/linear+span">linear span</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> inside <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> is all of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>.</p> <p>The vector space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> is called <em>finitely generated</em> if it admits a generating set (spanning set) which is a <a class="existingWikiWord" href="/nlab/show/finite+set">finite set</a>.</p> </div> </p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/orthogonal+basis">orthogonal basis</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/basis+in+functional+analysis">basis in functional analysis</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Schauder+basis">Schauder basis</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dual+basis">dual basis</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/basis">basis</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mutually+unbiased+bases">mutually unbiased bases</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Gelfand-Tsetlin+basis">Gelfand-Tsetlin basis</a> (in <a class="existingWikiWord" href="/nlab/show/representation+theory">representation theory</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/seminormal+basis">seminormal basis</a> (in <a class="existingWikiWord" href="/nlab/show/representation+theory+of+the+symmetric+group">representation theory of the symmetric group</a>)</p> </li> </ul> <h2 id="references">References</h2> <p>Lecture notes with much conceptual exposition:</p> <ul> <li>Karen E Smith, <em>Bases for infinite-dimensional vector spaces</em> [<a href="https://dept.math.lsa.umich.edu/~kesmith/infinite.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/Smith-InfiniteDimBases.pdf" title="pdf">pdf</a>]</li> </ul> <p>Lecture notes with the proofs concisely spelled out:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Christoph+Schweigert">Christoph Schweigert</a>, <em>Basis und Dimension</em>, §2.4 in: <em>Lineare Algebra</em>, lecture notes, Hamburg (2022) [<a href="https://www.math.uni-hamburg.de/home/schweigert/skripten/laskript.pdf">pdf</a>]</li> </ul> <p>See also:</p> <ul> <li>Wikipedia, <em><a href="https://en.wikipedia.org/wiki/Basis_(linear_algebra)">Basis (linear algebra)</a></em></li> </ul> <p>The original discussion (for <a class="existingWikiWord" href="/nlab/show/real+numbers"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mi>ℝ</mi> </mrow> <annotation encoding="application/x-tex">\mathbb{R}</annotation> </semantics> </math></a> regarded as a <a class="existingWikiWord" href="/nlab/show/rational+vector+space">rational vector space</a>) after which <em>Hamel bases</em> are named:</p> <ul> <li id="Hamel1905"><a class="existingWikiWord" href="/nlab/show/Georg+Hamel">Georg Hamel</a>, pp. 460 of: <em>Eine Basis aller Zahlen und die unstetigen Lösungen der Funktionalgleichung: <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>+</mo><mi>f</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x + y) = f(x) + f(y)</annotation></semantics></math></em>, Mathematische Annalen <strong>60</strong> (1905) 459–462 [<a href="https://doi.org/10.1007/BF01457624">doi:10.1007/BF01457624</a>]</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on August 26, 2023 at 17:24:41. 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