User Tools

Site Tools


0150--b-t-ng-th-c-h-lder-la-gi

Differences

This shows you the differences between two versions of the page.

Link to this comparison view

0150--b-t-ng-th-c-h-lder-la-gi [2018/11/07 17:09] (current)
Line 1: Line 1:
 +<​HTML>​ <​br><​div id="​mw-content-text"​ lang="​vi"​ dir="​ltr"><​div class="​mw-parser-output"><​p>​Trong giải tích toán học, <​b>​bất đẳng thức Hölder</​b>,​ đặt theo tên nhà toán họcĐức Otto Hölder, là một bất đẳng thức cơ bản liên quan đến các  không gian L<​sup><​i>​p</​i></​sup>:​ giả sử  <​i>​S</​i>​ là một không gian đo, với  1 ≤ <​i>​p</​i>,​ <​i>​q</​i>​ ≤ ∞ thỏa 1/<​i>​p</​i>​ + 1/<​i>​q</​i>​ = 1, đồng thời ​ <​i>​f</​i>​ thuộc ​ L<​sup><​i>​p</​i></​sup>​(<​i>​S</​i>​) và <​i>​g</​i>​ thuộc L<​sup><​i>​q</​i></​sup>​(<​i>​S</​i>​). Khi đó  <​i>​fg</​i>​ thuộc ​ L<​sup>​1</​sup>​(<​i>​S</​i>​) và
 +</p>
 +<​dl><​dd><​span class="​mwe-math-element"><​span class="​mwe-math-mathml-inline mwe-math-mathml-a11y"​ style="​display:​ none;"><​math xmlns="​http://​www.w3.org/​1998/​Math/​MathML"​ alttext="​{displaystyle |fg|_{1}leq |f|_{p}|g|_{q}.}"><​semantics><​mrow class="​MJX-TeXAtom-ORD"><​mstyle displaystyle="​true"​ scriptlevel="​0"><​mo fence="​false"​ stretchy="​false">​‖<​!-- &#8214; --></​mo><​mi>​f</​mi><​mi>​g</​mi><​msub><​mo fence="​false"​ stretchy="​false">​‖<​!-- &#8214; --></​mo><​mrow class="​MJX-TeXAtom-ORD"><​mn>​1</​mn></​mrow></​msub><​mo>​≤<​!-- &le; --></​mo><​mo fence="​false"​ stretchy="​false">​‖<​!-- &#8214; --></​mo><​mi>​f</​mi><​msub><​mo fence="​false"​ stretchy="​false">​‖<​!-- &#8214; --></​mo><​mrow class="​MJX-TeXAtom-ORD"><​mi>​p</​mi></​mrow></​msub><​mo fence="​false"​ stretchy="​false">​‖<​!-- &#8214; --></​mo><​mi>​g</​mi><​msub><​mo fence="​false"​ stretchy="​false">​‖<​!-- &#8214; --></​mo><​mrow class="​MJX-TeXAtom-ORD"><​mi>​q</​mi></​mrow></​msub><​mo>​.</​mo></​mstyle></​mrow><​annotation encoding="​application/​x-tex">​{displaystyle |fg|_{1}leq |f|_{p}|g|_{q}.}</​annotation></​semantics></​math></​span><​img src="​https://​wikimedia.org/​api/​rest_v1/​media/​math/​render/​svg/​fcc6ef64b19122c3944cfee255b4f7cbd9f9afd9"​ class="​mwe-math-fallback-image-inline"​ aria-hidden="​true"​ style="​vertical-align:​ -1.005ex; width:​18.611ex;​ height:​3.009ex;"​ alt="​{displaystyle |fg|_{1}leq |f|_{p}|g|_{q}.}"/></​span></​dd></​dl><​p>​Các số <​i>​p</​i>​ và <​i>​q</​i>​ nói trên được gọi là <​b>​liên hợp Holder</​b>​ của lẫn nhau.
 +</​p><​p>​Bất đẳng thức Holder được dùng để chứng minh bất đẳng thức tam giác tổng quát trong không gian L<​sup><​i>​p</​i></​sup>, ​ bất đẳng thức Minkowski và cũng dùng để chứng minh L<​sup><​i>​p</​i></​sup>​ là đối ngẫu với L<​sup><​i>​q</​i></​sup>​.
 +</p>
  
 +
 +<​h2><​span id="​C.C3.A1c_tr.C6.B0.E1.BB.9Dng_h.E1.BB.A3p_.C4.91.E1.BA.B7c_bi.E1.BB.87t_.C4.91.C3.A1ng_ch.C3.BA_.C3.BD"/><​span class="​mw-headline"​ id="​Các_trường_hợp_đặc_biệt_đáng_chú_ý">​Các trường hợp đặc biệt đáng chú ý</​span><​span class="​mw-editsection"><​span class="​mw-editsection-bracket">​[</​span>​sửa<​span class="​mw-editsection-divider">​ | </​span>​sửa mã nguồn<​span class="​mw-editsection-bracket">​]</​span></​span></​h2>​
 +<​dl><​dd><​span class="​mwe-math-element"><​span class="​mwe-math-mathml-inline mwe-math-mathml-a11y"​ style="​display:​ none;"><​math xmlns="​http://​www.w3.org/​1998/​Math/​MathML"​ alttext="​{displaystyle sum _{k=1}^{n}|x_{k}y_{k}|leq left(sum _{k=1}^{n}|x_{k}|^{p}right)^{1/​p}left(sum _{k=1}^{n}|y_{k}|^{q}right)^{1/​q}}"><​semantics><​mrow class="​MJX-TeXAtom-ORD"><​mstyle displaystyle="​true"​ scriptlevel="​0"><​munderover><​mo>​∑<​!-- &sum; --></​mo><​mrow class="​MJX-TeXAtom-ORD"><​mi>​k</​mi><​mo>​=</​mo><​mn>​1</​mn></​mrow><​mrow class="​MJX-TeXAtom-ORD"><​mi>​n</​mi></​mrow></​munderover><​mrow class="​MJX-TeXAtom-ORD"><​mo stretchy="​false">​|</​mo></​mrow><​msub><​mi>​x</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mi>​k</​mi></​mrow></​msub><​msub><​mi>​y</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mi>​k</​mi></​mrow></​msub><​mrow class="​MJX-TeXAtom-ORD"><​mo stretchy="​false">​|</​mo></​mrow><​mo>​≤<​!-- &le; --></​mo><​msup><​mrow><​mo>​(</​mo><​mrow><​munderover><​mo>​∑<​!-- &sum; --></​mo><​mrow class="​MJX-TeXAtom-ORD"><​mi>​k</​mi><​mo>​=</​mo><​mn>​1</​mn></​mrow><​mrow class="​MJX-TeXAtom-ORD"><​mi>​n</​mi></​mrow></​munderover><​mrow class="​MJX-TeXAtom-ORD"><​mo stretchy="​false">​|</​mo></​mrow><​msub><​mi>​x</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mi>​k</​mi></​mrow></​msub><​msup><​mrow class="​MJX-TeXAtom-ORD"><​mo stretchy="​false">​|</​mo></​mrow><​mrow class="​MJX-TeXAtom-ORD"><​mi>​p</​mi></​mrow></​msup></​mrow><​mo>​)</​mo></​mrow><​mrow class="​MJX-TeXAtom-ORD"><​mn>​1</​mn><​mrow class="​MJX-TeXAtom-ORD"><​mo>/</​mo></​mrow><​mi>​p</​mi></​mrow></​msup><​msup><​mrow><​mo>​(</​mo><​mrow><​munderover><​mo>​∑<​!-- &sum; --></​mo><​mrow class="​MJX-TeXAtom-ORD"><​mi>​k</​mi><​mo>​=</​mo><​mn>​1</​mn></​mrow><​mrow class="​MJX-TeXAtom-ORD"><​mi>​n</​mi></​mrow></​munderover><​mrow class="​MJX-TeXAtom-ORD"><​mo stretchy="​false">​|</​mo></​mrow><​msub><​mi>​y</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mi>​k</​mi></​mrow></​msub><​msup><​mrow class="​MJX-TeXAtom-ORD"><​mo stretchy="​false">​|</​mo></​mrow><​mrow class="​MJX-TeXAtom-ORD"><​mi>​q</​mi></​mrow></​msup></​mrow><​mo>​)</​mo></​mrow><​mrow class="​MJX-TeXAtom-ORD"><​mn>​1</​mn><​mrow class="​MJX-TeXAtom-ORD"><​mo>/</​mo></​mrow><​mi>​q</​mi></​mrow></​msup></​mstyle></​mrow>​{displaystyle sum _{k=1}^{n}|x_{k}y_{k}|leq left(sum _{k=1}^{n}|x_{k}|^{p}right)^{1/​p}left(sum _{k=1}^{n}|y_{k}|^{q}right)^{1/​q}}</​annotation></​semantics></​math></​span><​img src="​https://​wikimedia.org/​api/​rest_v1/​media/​math/​render/​svg/​42233a6c7f2c684336ff7294d734474f8dec203c"​ class="​mwe-math-fallback-image-inline"​ aria-hidden="​true"​ style="​vertical-align:​ -3.171ex; width:​42.244ex;​ height:​8.009ex;"​ alt="​{displaystyle sum _{k=1}^{n}|x_{k}y_{k}|leq left(sum _{k=1}^{n}|x_{k}|^{p}right)^{1/​p}left(sum _{k=1}^{n}|y_{k}|^{q}right)^{1/​q}}"/></​span></​dd></​dl><​dl><​dd><​span class="​mwe-math-element"><​span class="​mwe-math-mathml-inline mwe-math-mathml-a11y"​ style="​display:​ none;"><​math xmlns="​http://​www.w3.org/​1998/​Math/​MathML"​ alttext="​{displaystyle sum limits _{n=1}^{infty }|x_{n}cdot y_{n}|leq left(sum limits _{n=1}^{infty }|x_{n}|^{p}right)^{1/​p}cdot left(sum limits _{n=1}^{infty }|y_{n}|^{q}right)^{1/​q},;​forall xin l^{p},yin l^{q}}"><​semantics><​mrow class="​MJX-TeXAtom-ORD"><​mstyle displaystyle="​true"​ scriptlevel="​0"><​munderover><​mo movablelimits="​false">​∑<​!-- &sum; --></​mo><​mrow class="​MJX-TeXAtom-ORD"><​mi>​n</​mi><​mo>​=</​mo><​mn>​1</​mn></​mrow><​mrow class="​MJX-TeXAtom-ORD"><​mi mathvariant="​normal">​∞<​!-- &infin; --></​mi></​mrow></​munderover><​mrow class="​MJX-TeXAtom-ORD"><​mo stretchy="​false">​|</​mo></​mrow><​msub><​mi>​x</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mi>​n</​mi></​mrow></​msub><​mo>​⋅<​!-- &sdot; --></​mo><​msub><​mi>​y</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mi>​n</​mi></​mrow></​msub><​mrow class="​MJX-TeXAtom-ORD"><​mo stretchy="​false">​|</​mo></​mrow><​mo>​≤<​!-- &le; --></​mo><​msup><​mrow><​mo>​(</​mo><​mrow><​munderover><​mo movablelimits="​false">​∑<​!-- &sum; --></​mo><​mrow class="​MJX-TeXAtom-ORD"><​mi>​n</​mi><​mo>​=</​mo><​mn>​1</​mn></​mrow><​mrow class="​MJX-TeXAtom-ORD"><​mi mathvariant="​normal">​∞<​!-- &infin; --></​mi></​mrow></​munderover><​mrow class="​MJX-TeXAtom-ORD"><​mo stretchy="​false">​|</​mo></​mrow><​msub><​mi>​x</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mi>​n</​mi></​mrow></​msub><​msup><​mrow class="​MJX-TeXAtom-ORD"><​mo stretchy="​false">​|</​mo></​mrow><​mrow class="​MJX-TeXAtom-ORD"><​mi>​p</​mi></​mrow></​msup></​mrow><​mo>​)</​mo></​mrow><​mrow class="​MJX-TeXAtom-ORD"><​mn>​1</​mn><​mrow class="​MJX-TeXAtom-ORD"><​mo>/</​mo></​mrow><​mi>​p</​mi></​mrow></​msup><​mo>​⋅<​!-- &sdot; --></​mo><​msup><​mrow><​mo>​(</​mo><​mrow><​munderover><​mo movablelimits="​false">​∑<​!-- &sum; --></​mo><​mrow class="​MJX-TeXAtom-ORD"><​mi>​n</​mi><​mo>​=</​mo><​mn>​1</​mn></​mrow><​mrow class="​MJX-TeXAtom-ORD"><​mi mathvariant="​normal">​∞<​!-- &infin; --></​mi></​mrow></​munderover><​mrow class="​MJX-TeXAtom-ORD"><​mo stretchy="​false">​|</​mo></​mrow><​msub><​mi>​y</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mi>​n</​mi></​mrow></​msub><​msup><​mrow class="​MJX-TeXAtom-ORD"><​mo stretchy="​false">​|</​mo></​mrow><​mrow class="​MJX-TeXAtom-ORD"><​mi>​q</​mi></​mrow></​msup></​mrow><​mo>​)</​mo></​mrow><​mrow class="​MJX-TeXAtom-ORD"><​mn>​1</​mn><​mrow class="​MJX-TeXAtom-ORD"><​mo>/</​mo></​mrow><​mi>​q</​mi></​mrow></​msup><​mo>,</​mo><​mspace width="​thickmathspace"/><​mi mathvariant="​normal">​∀<​!-- &​forall;​ --></​mi><​mi>​x</​mi><​mo>​∈<​!-- &isin; --></​mo><​msup><​mi>​l</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mi>​p</​mi></​mrow></​msup><​mo>,</​mo><​mi>​y</​mi><​mo>​∈<​!-- &isin; --></​mo><​msup><​mi>​l</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mi>​q</​mi></​mrow></​msup></​mstyle></​mrow><​annotation encoding="​application/​x-tex">​{displaystyle sum limits _{n=1}^{infty }|x_{n}cdot y_{n}|leq left(sum limits _{n=1}^{infty }|x_{n}|^{p}right)^{1/​p}cdot left(sum limits _{n=1}^{infty }|y_{n}|^{q}right)^{1/​q},;​forall xin l^{p},yin l^{q}}</​annotation></​semantics></​math></​span><​img src="​https://​wikimedia.org/​api/​rest_v1/​media/​math/​render/​svg/​f1a43549299dbcb4236e5cbd3ae07829a8204c77"​ class="​mwe-math-fallback-image-inline"​ aria-hidden="​true"​ style="​vertical-align:​ -3.171ex; width:​61.728ex;​ height:​8.009ex;"​ alt="​{displaystyle sum limits _{n=1}^{infty }|x_{n}cdot y_{n}|leq left(sum limits _{n=1}^{infty }|x_{n}|^{p}right)^{1/​p}cdot left(sum limits _{n=1}^{infty }|y_{n}|^{q}right)^{1/​q},;​forall xin l^{p},yin l^{q}}"/></​span>​.</​dd></​dl><​ul><​li>​Trong trường hợp không gian của các <a href="​http://​vi.wikipedia.org/​w/​index.php?​title=H%C3%A0m(to%C3%A1n)&​amp;​action=edit&​amp;​redlink=1"​ class="​new"​ title="​Hàm(toán) (trang chưa được viết)">​hàm giá trị phức khả tích, chúng ta có</​li></​ul><​dl><​dd><​span class="​mwe-math-element"><​span class="​mwe-math-mathml-inline mwe-math-mathml-a11y"​ style="​display:​ none;"><​math xmlns="​http://​www.w3.org/​1998/​Math/​MathML"​ alttext="​{displaystyle left|int f(x)g(x),​dxright|leq left(int left|f(x)right|^{p},​dxright)^{1/​p}cdot left(int left|g(x)right|^{q},​dxright)^{1/​q}.}"><​semantics><​mrow class="​MJX-TeXAtom-ORD"><​mstyle displaystyle="​true"​ scriptlevel="​0"><​mrow><​mo>​|</​mo><​mrow><​mo>​∫<​!-- &int; --></​mo><​mi>​f</​mi><​mo stretchy="​false">​(</​mo><​mi>​x</​mi><​mo stretchy="​false">​)</​mo><​mi>​g</​mi><​mo stretchy="​false">​(</​mo><​mi>​x</​mi><​mo stretchy="​false">​)</​mo><​mspace width="​thinmathspace"/><​mi>​d</​mi><​mi>​x</​mi></​mrow><​mo>​|</​mo></​mrow><​mo>​≤<​!-- &le; --></​mo><​msup><​mrow><​mo>​(</​mo><​mrow><​mo>​∫<​!-- &int; --></​mo><​msup><​mrow><​mo>​|</​mo><​mrow><​mi>​f</​mi><​mo stretchy="​false">​(</​mo><​mi>​x</​mi><​mo stretchy="​false">​)</​mo></​mrow><​mo>​|</​mo></​mrow><​mrow class="​MJX-TeXAtom-ORD"><​mi>​p</​mi></​mrow></​msup><​mspace width="​thinmathspace"/><​mi>​d</​mi><​mi>​x</​mi></​mrow><​mo>​)</​mo></​mrow><​mrow class="​MJX-TeXAtom-ORD"><​mn>​1</​mn><​mrow class="​MJX-TeXAtom-ORD"><​mo>/</​mo></​mrow><​mi>​p</​mi></​mrow></​msup><​mo>​⋅<​!-- &sdot; --></​mo><​msup><​mrow><​mo>​(</​mo><​mrow><​mo>​∫<​!-- &int; --></​mo><​msup><​mrow><​mo>​|</​mo><​mrow><​mi>​g</​mi><​mo stretchy="​false">​(</​mo><​mi>​x</​mi><​mo stretchy="​false">​)</​mo></​mrow><​mo>​|</​mo></​mrow><​mrow class="​MJX-TeXAtom-ORD"><​mi>​q</​mi></​mrow></​msup><​mspace width="​thinmathspace"/><​mi>​d</​mi><​mi>​x</​mi></​mrow><​mo>​)</​mo></​mrow><​mrow class="​MJX-TeXAtom-ORD"><​mn>​1</​mn><​mrow class="​MJX-TeXAtom-ORD"><​mo>/</​mo></​mrow><​mi>​q</​mi></​mrow></​msup><​mo>​.</​mo></​mstyle></​mrow><​annotation encoding="​application/​x-tex">​{displaystyle left|int f(x)g(x),​dxright|leq left(int left|f(x)right|^{p},​dxright)^{1/​p}cdot left(int left|g(x)right|^{q},​dxright)^{1/​q}.}</​annotation></​semantics></​math></​span><​img src="​https://​wikimedia.org/​api/​rest_v1/​media/​math/​render/​svg/​4f2cbf59ce9c92619af4bc0c1ceaeb083f4429c9"​ class="​mwe-math-fallback-image-inline"​ aria-hidden="​true"​ style="​vertical-align:​ -2.505ex; width:​57.417ex;​ height:​6.676ex;"​ alt="​{displaystyle left|int f(x)g(x),​dxright|leq left(int left|f(x)right|^{p},​dxright)^{1/​p}cdot left(int left|g(x)right|^{q},​dxright)^{1/​q}.}"/></​span></​dd></​dl><​ul><​li>​Dạng đại số thường gặp trong chứng minh bất đẳng thức của bất đẳng thức Holder</​li></​ul><​p><​span class="​mwe-math-element"><​span class="​mwe-math-mathml-inline mwe-math-mathml-a11y"​ style="​display:​ none;"><​math xmlns="​http://​www.w3.org/​1998/​Math/​MathML"​ alttext="​{displaystyle coprod _{i=1}^{m}{biggl (}sum _{j=1}^{n}a_{i,​j}{biggr )}geq {biggl (}sum _{j=1}^{n}{sqrt[{m}]{prod _{i=1}^{m}a_{i,​j}}}{biggr )}^{m}}"><​semantics><​mrow class="​MJX-TeXAtom-ORD"><​mstyle displaystyle="​true"​ scriptlevel="​0"><​munderover><​mo>​∐<​!-- &#8720; --></​mo><​mrow class="​MJX-TeXAtom-ORD"><​mi>​i</​mi><​mo>​=</​mo><​mn>​1</​mn></​mrow><​mrow class="​MJX-TeXAtom-ORD"><​mi>​m</​mi></​mrow></​munderover><​mrow class="​MJX-TeXAtom-ORD"><​mrow class="​MJX-TeXAtom-OPEN"><​mo maxsize="​2.047em"​ minsize="​2.047em">​(</​mo></​mrow></​mrow><​munderover><​mo>​∑<​!-- &sum; --></​mo><​mrow class="​MJX-TeXAtom-ORD"><​mi>​j</​mi><​mo>​=</​mo><​mn>​1</​mn></​mrow><​mrow class="​MJX-TeXAtom-ORD"><​mi>​n</​mi></​mrow></​munderover><​msub><​mi>​a</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mi>​i</​mi><​mo>,</​mo><​mi>​j</​mi></​mrow></​msub><​mrow class="​MJX-TeXAtom-ORD"><​mrow class="​MJX-TeXAtom-CLOSE"><​mo maxsize="​2.047em"​ minsize="​2.047em">​)</​mo></​mrow></​mrow><​mo>​≥<​!-- &ge; --></​mo><​mrow class="​MJX-TeXAtom-ORD"><​mrow class="​MJX-TeXAtom-OPEN"><​mo maxsize="​2.047em"​ minsize="​2.047em">​(</​mo></​mrow></​mrow><​munderover><​mo>​∑<​!-- &sum; --></​mo><​mrow class="​MJX-TeXAtom-ORD"><​mi>​j</​mi><​mo>​=</​mo><​mn>​1</​mn></​mrow><​mrow class="​MJX-TeXAtom-ORD"><​mi>​n</​mi></​mrow></​munderover><​mrow class="​MJX-TeXAtom-ORD"><​mroot><​mrow><​munderover><​mo>​∏<​!-- &prod; --></​mo><​mrow class="​MJX-TeXAtom-ORD"><​mi>​i</​mi><​mo>​=</​mo><​mn>​1</​mn></​mrow><​mrow class="​MJX-TeXAtom-ORD"><​mi>​m</​mi></​mrow></​munderover><​msub><​mi>​a</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mi>​i</​mi><​mo>,</​mo><​mi>​j</​mi></​mrow></​msub></​mrow><​mrow class="​MJX-TeXAtom-ORD"><​mi>​m</​mi></​mrow></​mroot></​mrow><​msup><​mrow class="​MJX-TeXAtom-ORD"><​mrow class="​MJX-TeXAtom-CLOSE"><​mo maxsize="​2.047em"​ minsize="​2.047em">​)</​mo></​mrow></​mrow><​mrow class="​MJX-TeXAtom-ORD"><​mi>​m</​mi></​mrow></​msup></​mstyle></​mrow><​annotation encoding="​application/​x-tex">​{displaystyle coprod _{i=1}^{m}{biggl (}sum _{j=1}^{n}a_{i,​j}{biggr )}geq {biggl (}sum _{j=1}^{n}{sqrt[{m}]{prod _{i=1}^{m}a_{i,​j}}}{biggr )}^{m}}</​annotation></​semantics></​math></​span><​img src="​https://​wikimedia.org/​api/​rest_v1/​media/​math/​render/​svg/​c9ef047f4a27b788841e2ff109be470b30e9c10f"​ class="​mwe-math-fallback-image-inline"​ aria-hidden="​true"​ style="​vertical-align:​ -3.338ex; width:​35.24ex;​ height:​7.509ex;"​ alt="​{displaystyle coprod _{i=1}^{m}{biggl (}sum _{j=1}^{n}a_{i,​j}{biggr )}geq {biggl (}sum _{j=1}^{n}{sqrt[{m}]{prod _{i=1}^{m}a_{i,​j}}}{biggr )}^{m}}"/></​span>​
 +</p>
 +<​ul><​li>​Trong trường hợp  không gian xác suất <span class="​mwe-math-element"><​span class="​mwe-math-mathml-inline mwe-math-mathml-a11y"​ style="​display:​ none;"><​math xmlns="​http://​www.w3.org/​1998/​Math/​MathML"​ alttext="​{displaystyle (Omega ,{mathcal {F}},mathbb {P} )}"><​semantics><​mrow class="​MJX-TeXAtom-ORD"><​mstyle displaystyle="​true"​ scriptlevel="​0"><​mo stretchy="​false">​(</​mo><​mi mathvariant="​normal">​Ω<​!-- &Omega; --></​mi><​mo>,</​mo><​mrow class="​MJX-TeXAtom-ORD"><​mrow class="​MJX-TeXAtom-ORD"><​mi class="​MJX-tex-caligraphic"​ mathvariant="​script">​F</​mi></​mrow></​mrow><​mo>,</​mo><​mrow class="​MJX-TeXAtom-ORD"><​mi mathvariant="​double-struck">​P</​mi></​mrow><​mo stretchy="​false">​)</​mo></​mstyle></​mrow>​{displaystyle (Omega ,{mathcal {F}},mathbb {P} )}</​annotation></​semantics></​math></​span><​img src="​https://​wikimedia.org/​api/​rest_v1/​media/​math/​render/​svg/​6bb8743f7565082ed1a9ee0490d9d71be82eafaa"​ class="​mwe-math-fallback-image-inline"​ aria-hidden="​true"​ style="​vertical-align:​ -0.838ex; width:​8.902ex;​ height:​2.843ex;"​ alt="​{displaystyle (Omega ,{mathcal {F}},mathbb {P} )}"/></​span>,​ <span class="​mwe-math-element"><​span class="​mwe-math-mathml-inline mwe-math-mathml-a11y"​ style="​display:​ none;"><​math xmlns="​http://​www.w3.org/​1998/​Math/​MathML"​ alttext="​{displaystyle L^{p}(Omega ,{mathcal {F}},mathbb {P} )}"><​semantics><​mrow class="​MJX-TeXAtom-ORD"><​mstyle displaystyle="​true"​ scriptlevel="​0"><​msup><​mi>​L</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mi>​p</​mi></​mrow></​msup><​mo stretchy="​false">​(</​mo><​mi mathvariant="​normal">​Ω<​!-- &Omega; --></​mi><​mo>,</​mo><​mrow class="​MJX-TeXAtom-ORD"><​mrow class="​MJX-TeXAtom-ORD"><​mi class="​MJX-tex-caligraphic"​ mathvariant="​script">​F</​mi></​mrow></​mrow><​mo>,</​mo><​mrow class="​MJX-TeXAtom-ORD"><​mi mathvariant="​double-struck">​P</​mi></​mrow><​mo stretchy="​false">​)</​mo></​mstyle></​mrow><​annotation encoding="​application/​x-tex">​{displaystyle L^{p}(Omega ,{mathcal {F}},mathbb {P} )}</​annotation></​semantics></​math></​span><​img src="​https://​wikimedia.org/​api/​rest_v1/​media/​math/​render/​svg/​74795e2f53c1e3473ff8d4f0d402aa9bc62d78ea"​ class="​mwe-math-fallback-image-inline"​ aria-hidden="​true"​ style="​vertical-align:​ -0.838ex; width:​11.544ex;​ height:​2.843ex;"​ alt="​{displaystyle L^{p}(Omega ,{mathcal {F}},mathbb {P} )}"/></​span>​ là các ký hiệu để chỉ không gian của các biến ngẫu nhiên với <a href="​http://​vi.wikipedia.org/​w/​index.php?​title=Moment(to%C3%A1n)&​amp;​action=edit&​amp;​redlink=1"​ class="​new"​ title="​Moment(toán) (trang chưa được viết)">​moment<​i>​p</​i>​ hữu hạn,</​li></​ul><​p><​span class="​mwe-math-element"><​span class="​mwe-math-mathml-inline mwe-math-mathml-a11y"​ style="​display:​ none;"><​math xmlns="​http://​www.w3.org/​1998/​Math/​MathML"​ alttext="​{displaystyle mathbb {E} left[|X|^{p}right]&​lt;​infty }"><​semantics><​mrow class="​MJX-TeXAtom-ORD"><​mstyle displaystyle="​true"​ scriptlevel="​0"><​mrow class="​MJX-TeXAtom-ORD"><​mi mathvariant="​double-struck">​E</​mi></​mrow><​mrow><​mo>​[</​mo><​mrow><​mrow class="​MJX-TeXAtom-ORD"><​mo stretchy="​false">​|</​mo></​mrow><​mi>​X</​mi><​msup><​mrow class="​MJX-TeXAtom-ORD"><​mo stretchy="​false">​|</​mo></​mrow><​mrow class="​MJX-TeXAtom-ORD"><​mi>​p</​mi></​mrow></​msup></​mrow><​mo>​]</​mo></​mrow><​mo>&​lt;</​mo><​mi mathvariant="​normal">​∞<​!-- &infin; --></​mi></​mstyle></​mrow>​{displaystyle mathbb {E} left[|X|^{p}right]&​lt;​infty }</​annotation></​semantics></​math></​span><​img src="​https://​wikimedia.org/​api/​rest_v1/​media/​math/​render/​svg/​062c1b0d0f1f9f14e2d4c91765f579fd753615a6"​ class="​mwe-math-fallback-image-inline"​ aria-hidden="​true"​ style="​vertical-align:​ -0.838ex; width:​12.986ex;​ height:​3.009ex;"​ alt="​{displaystyle mathbb {E} left[|X|^{p}right]&​lt;​infty }"/></​span>,​ trong đó  <span class="​mwe-math-element"><​span class="​mwe-math-mathml-inline mwe-math-mathml-a11y"​ style="​display:​ none;"><​math xmlns="​http://​www.w3.org/​1998/​Math/​MathML"​ alttext="​{displaystyle mathbb {E} }"><​semantics><​mrow class="​MJX-TeXAtom-ORD"><​mstyle displaystyle="​true"​ scriptlevel="​0"><​mrow class="​MJX-TeXAtom-ORD"><​mi mathvariant="​double-struck">​E</​mi></​mrow></​mstyle></​mrow><​annotation encoding="​application/​x-tex">​{displaystyle mathbb {E} }</​annotation></​semantics></​math></​span><​img src="​https://​wikimedia.org/​api/​rest_v1/​media/​math/​render/​svg/​ad9faf1fd4a61d36d7f8a2f3204f3805a43c0d4a"​ class="​mwe-math-fallback-image-inline"​ aria-hidden="​true"​ style="​vertical-align:​ -0.338ex; width:​1.55ex;​ height:​2.176ex;"​ alt="​{displaystyle mathbb {E} }"/></​span>​ là ký hiệu chỉ  <a href="​http://​vi.wikipedia.org/​wiki/​Gi%C3%A1_tr%E1%BB%8B_k%E1%BB%B3_v%E1%BB%8Dng"​ title="​Giá trị kỳ vọng">​giá trị kỳ vọng. Bất đẳng thức Holder trở thành
 +</p>
 +<​dl><​dd><​span class="​mwe-math-element"><​span class="​mwe-math-mathml-inline mwe-math-mathml-a11y"​ style="​display:​ none;"><​math xmlns="​http://​www.w3.org/​1998/​Math/​MathML"​ alttext="​{displaystyle mathbb {E} |XY|leq left(mathbb {E} |X|^{p}right)^{1/​p}cdot left(mathbb {E} |Y|^{q}right)^{1/​q},;​forall Xin L^{p},Yin L^{q}}"><​semantics><​mrow class="​MJX-TeXAtom-ORD"><​mstyle displaystyle="​true"​ scriptlevel="​0"><​mrow class="​MJX-TeXAtom-ORD"><​mi mathvariant="​double-struck">​E</​mi></​mrow><​mrow class="​MJX-TeXAtom-ORD"><​mo stretchy="​false">​|</​mo></​mrow><​mi>​X</​mi><​mi>​Y</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mo stretchy="​false">​|</​mo></​mrow><​mo>​≤<​!-- &le; --></​mo><​msup><​mrow><​mo>​(</​mo><​mrow><​mrow class="​MJX-TeXAtom-ORD"><​mi mathvariant="​double-struck">​E</​mi></​mrow><​mrow class="​MJX-TeXAtom-ORD"><​mo stretchy="​false">​|</​mo></​mrow><​mi>​X</​mi><​msup><​mrow class="​MJX-TeXAtom-ORD"><​mo stretchy="​false">​|</​mo></​mrow><​mrow class="​MJX-TeXAtom-ORD"><​mi>​p</​mi></​mrow></​msup></​mrow><​mo>​)</​mo></​mrow><​mrow class="​MJX-TeXAtom-ORD"><​mn>​1</​mn><​mrow class="​MJX-TeXAtom-ORD"><​mo>/</​mo></​mrow><​mi>​p</​mi></​mrow></​msup><​mo>​⋅<​!-- &sdot; --></​mo><​msup><​mrow><​mo>​(</​mo><​mrow><​mrow class="​MJX-TeXAtom-ORD"><​mi mathvariant="​double-struck">​E</​mi></​mrow><​mrow class="​MJX-TeXAtom-ORD"><​mo stretchy="​false">​|</​mo></​mrow><​mi>​Y</​mi><​msup><​mrow class="​MJX-TeXAtom-ORD"><​mo stretchy="​false">​|</​mo></​mrow><​mrow class="​MJX-TeXAtom-ORD"><​mi>​q</​mi></​mrow></​msup></​mrow><​mo>​)</​mo></​mrow><​mrow class="​MJX-TeXAtom-ORD"><​mn>​1</​mn><​mrow class="​MJX-TeXAtom-ORD"><​mo>/</​mo></​mrow><​mi>​q</​mi></​mrow></​msup><​mo>,</​mo><​mspace width="​thickmathspace"/><​mi mathvariant="​normal">​∀<​!-- &​forall;​ --></​mi><​mi>​X</​mi><​mo>​∈<​!-- &isin; --></​mo><​msup><​mi>​L</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mi>​p</​mi></​mrow></​msup><​mo>,</​mo><​mi>​Y</​mi><​mo>​∈<​!-- &isin; --></​mo><​msup><​mi>​L</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mi>​q</​mi></​mrow></​msup></​mstyle></​mrow><​annotation encoding="​application/​x-tex">​{displaystyle mathbb {E} |XY|leq left(mathbb {E} |X|^{p}right)^{1/​p}cdot left(mathbb {E} |Y|^{q}right)^{1/​q},;​forall Xin L^{p},Yin L^{q}}</​annotation></​semantics></​math></​span><​img src="​https://​wikimedia.org/​api/​rest_v1/​media/​math/​render/​svg/​647e5afb2cd4f783502f51e79eeb610ac2b0196b"​ class="​mwe-math-fallback-image-inline"​ aria-hidden="​true"​ style="​vertical-align:​ -0.838ex; width:​50.472ex;​ height:​3.509ex;"​ alt="​{displaystyle mathbb {E} |XY|leq left(mathbb {E} |X|^{p}right)^{1/​p}cdot left(mathbb {E} |Y|^{q}right)^{1/​q},;​forall Xin L^{p},Yin L^{q}}"/></​span>​.</​dd></​dl>​
 +<​p>​Có thể chứng minh trường hợp tổng quát sau bằng phương pháp quy nạp
 +</​p><​p>​Giả sử  <span class="​mwe-math-element"><​span class="​mwe-math-mathml-inline mwe-math-mathml-a11y"​ style="​display:​ none;"><​math xmlns="​http://​www.w3.org/​1998/​Math/​MathML"​ alttext="​{displaystyle p_{k}geq 1,k=1,ldots n}"><​semantics><​mrow class="​MJX-TeXAtom-ORD"><​mstyle displaystyle="​true"​ scriptlevel="​0"><​msub><​mi>​p</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mi>​k</​mi></​mrow></​msub><​mo>​≥<​!-- &ge; --></​mo><​mn>​1</​mn><​mo>,</​mo><​mi>​k</​mi><​mo>​=</​mo><​mn>​1</​mn><​mo>,</​mo><​mo>​…<​!-- &​hellip;​ --></​mo><​mi>​n</​mi></​mstyle></​mrow><​annotation encoding="​application/​x-tex">​{displaystyle p_{k}geq 1,k=1,ldots n}</​annotation></​semantics></​math></​span><​img src="​https://​wikimedia.org/​api/​rest_v1/​media/​math/​render/​svg/​f98b27a3ad0da081c6175ca555d1cffec4e0314d"​ class="​mwe-math-fallback-image-inline"​ aria-hidden="​true"​ style="​vertical-align:​ -0.671ex; margin-left:​ -0.089ex; width:​18.654ex;​ height:​2.509ex;"​ alt="​{displaystyle p_{k}geq 1,k=1,ldots n}"/></​span>​ sao cho 
 +</p>
 +<​dl><​dd><​span class="​mwe-math-element"><​span class="​mwe-math-mathml-inline mwe-math-mathml-a11y"​ style="​display:​ none;"><​math xmlns="​http://​www.w3.org/​1998/​Math/​MathML"​ alttext="​{displaystyle sum _{k=1}^{n}{frac {1}{p_{k}}}=1}"><​semantics><​mrow class="​MJX-TeXAtom-ORD"><​mstyle displaystyle="​true"​ scriptlevel="​0"><​munderover><​mo>​∑<​!-- &sum; --></​mo><​mrow class="​MJX-TeXAtom-ORD"><​mi>​k</​mi><​mo>​=</​mo><​mn>​1</​mn></​mrow><​mrow class="​MJX-TeXAtom-ORD"><​mi>​n</​mi></​mrow></​munderover><​mrow class="​MJX-TeXAtom-ORD"><​mfrac><​mn>​1</​mn><​msub><​mi>​p</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mi>​k</​mi></​mrow></​msub></​mfrac></​mrow><​mo>​=</​mo><​mn>​1</​mn></​mstyle></​mrow><​annotation encoding="​application/​x-tex">​{displaystyle sum _{k=1}^{n}{frac {1}{p_{k}}}=1}</​annotation></​semantics></​math></​span><​img src="​https://​wikimedia.org/​api/​rest_v1/​media/​math/​render/​svg/​4f2d78ee91279f4f9ae6a7484239b243150cd51f"​ class="​mwe-math-fallback-image-inline"​ aria-hidden="​true"​ style="​vertical-align:​ -3.005ex; width:​11.097ex;​ height:​6.843ex;"​ alt="​{displaystyle sum _{k=1}^{n}{frac {1}{p_{k}}}=1}"/></​span></​dd></​dl><​p>​Giả sử  <span class="​mwe-math-element"><​span class="​mwe-math-mathml-inline mwe-math-mathml-a11y"​ style="​display:​ none;"><​math xmlns="​http://​www.w3.org/​1998/​Math/​MathML"​ alttext="​{displaystyle u_{k}in L^{p_{k}}(S)}"><​semantics><​mrow class="​MJX-TeXAtom-ORD"><​mstyle displaystyle="​true"​ scriptlevel="​0"><​msub><​mi>​u</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mi>​k</​mi></​mrow></​msub><​mo>​∈<​!-- &isin; --></​mo><​msup><​mi>​L</​mi><​mrow class="​MJX-TeXAtom-ORD"><​msub><​mi>​p</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mi>​k</​mi></​mrow></​msub></​mrow></​msup><​mo stretchy="​false">​(</​mo><​mi>​S</​mi><​mo stretchy="​false">​)</​mo></​mstyle></​mrow><​annotation encoding="​application/​x-tex">​{displaystyle u_{k}in L^{p_{k}}(S)}</​annotation></​semantics></​math></​span><​img src="​https://​wikimedia.org/​api/​rest_v1/​media/​math/​render/​svg/​21c34571b8ae4267f1401c5f42d82857b2b16c94"​ class="​mwe-math-fallback-image-inline"​ aria-hidden="​true"​ style="​vertical-align:​ -0.838ex; width:​12.069ex;​ height:​2.843ex;"​ alt="​{displaystyle u_{k}in L^{p_{k}}(S)}"/></​span>​. Khi đó ta có <span class="​mwe-math-element"><​span class="​mwe-math-mathml-inline mwe-math-mathml-a11y"​ style="​display:​ none;"><​math xmlns="​http://​www.w3.org/​1998/​Math/​MathML"​ alttext="​{displaystyle prod _{k=1}^{n}u_{k}in L^{1}(S)}"><​semantics><​mrow class="​MJX-TeXAtom-ORD"><​mstyle displaystyle="​true"​ scriptlevel="​0"><​munderover><​mo>​∏<​!-- &prod; --></​mo><​mrow class="​MJX-TeXAtom-ORD"><​mi>​k</​mi><​mo>​=</​mo><​mn>​1</​mn></​mrow><​mrow class="​MJX-TeXAtom-ORD"><​mi>​n</​mi></​mrow></​munderover><​msub><​mi>​u</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mi>​k</​mi></​mrow></​msub><​mo>​∈<​!-- &isin; --></​mo><​msup><​mi>​L</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mn>​1</​mn></​mrow></​msup><​mo stretchy="​false">​(</​mo><​mi>​S</​mi><​mo stretchy="​false">​)</​mo></​mstyle></​mrow><​annotation encoding="​application/​x-tex">​{displaystyle prod _{k=1}^{n}u_{k}in L^{1}(S)}</​annotation></​semantics></​math></​span><​img src="​https://​wikimedia.org/​api/​rest_v1/​media/​math/​render/​svg/​27b5f94b39a8fe6aeefbe335cefd36720e1c5950"​ class="​mwe-math-fallback-image-inline"​ aria-hidden="​true"​ style="​vertical-align:​ -3.005ex; width:​14.561ex;​ height:​6.843ex;"​ alt="​{displaystyle prod _{k=1}^{n}u_{k}in L^{1}(S)}"/></​span> ​ và
 +</p>
 +<​dl><​dd><​span class="​mwe-math-element"><​span class="​mwe-math-mathml-inline mwe-math-mathml-a11y"​ style="​display:​ none;"><​math xmlns="​http://​www.w3.org/​1998/​Math/​MathML"​ alttext="​{displaystyle left|prod _{k=1}^{n}u_{k}right|_{displaystyle L^{1}(S)}leq prod _{k=1}^{n}|u_{k}|_{displaystyle L^{p_{k}}(S)}}"><​semantics><​mrow class="​MJX-TeXAtom-ORD"><​mstyle displaystyle="​true"​ scriptlevel="​0"><​msub><​mrow><​mo symmetric="​true">​‖</​mo><​mrow><​munderover><​mo>​∏<​!-- &prod; --></​mo><​mrow class="​MJX-TeXAtom-ORD"><​mi>​k</​mi><​mo>​=</​mo><​mn>​1</​mn></​mrow><​mrow class="​MJX-TeXAtom-ORD"><​mi>​n</​mi></​mrow></​munderover><​msub><​mi>​u</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mi>​k</​mi></​mrow></​msub></​mrow><​mo symmetric="​true">​‖</​mo></​mrow><​mrow class="​MJX-TeXAtom-ORD"><​mstyle displaystyle="​true"​ scriptlevel="​0"><​msup><​mi>​L</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mn>​1</​mn></​mrow></​msup><​mo stretchy="​false">​(</​mo><​mi>​S</​mi><​mo stretchy="​false">​)</​mo></​mstyle></​mrow></​msub><​mo>​≤<​!-- &le; --></​mo><​munderover><​mo>​∏<​!-- &prod; --></​mo><​mrow class="​MJX-TeXAtom-ORD"><​mi>​k</​mi><​mo>​=</​mo><​mn>​1</​mn></​mrow><​mrow class="​MJX-TeXAtom-ORD"><​mi>​n</​mi></​mrow></​munderover><​mo fence="​false"​ stretchy="​false">​‖<​!-- &#8214; --></​mo><​msub><​mi>​u</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mi>​k</​mi></​mrow></​msub><​msub><​mo fence="​false"​ stretchy="​false">​‖<​!-- &#8214; --></​mo><​mrow class="​MJX-TeXAtom-ORD"><​mstyle displaystyle="​true"​ scriptlevel="​0"><​msup><​mi>​L</​mi><​mrow class="​MJX-TeXAtom-ORD"><​msub><​mi>​p</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mi>​k</​mi></​mrow></​msub></​mrow></​msup><​mo stretchy="​false">​(</​mo><​mi>​S</​mi><​mo stretchy="​false">​)</​mo></​mstyle></​mrow></​msub></​mstyle></​mrow><​annotation encoding="​application/​x-tex">​{displaystyle left|prod _{k=1}^{n}u_{k}right|_{displaystyle L^{1}(S)}leq prod _{k=1}^{n}|u_{k}|_{displaystyle L^{p_{k}}(S)}}</​annotation></​semantics></​math></​span><​img src="​https://​wikimedia.org/​api/​rest_v1/​media/​math/​render/​svg/​83a18db86e8243f0de982bd7922838fd762485e0"​ class="​mwe-math-fallback-image-inline"​ aria-hidden="​true"​ style="​vertical-align:​ -3.671ex; width:​32.518ex;​ height:​7.843ex;"​ alt="​{displaystyle left|prod _{k=1}^{n}u_{k}right|_{displaystyle L^{1}(S)}leq prod _{k=1}^{n}|u_{k}|_{displaystyle L^{p_{k}}(S)}}"/></​span></​dd></​dl>​
 +
 +
 +
 +<​!-- ​
 +NewPP limit report
 +Parsed by mw1230
 +Cached time: 20181011092024
 +Cache expiry: 1900800
 +Dynamic content: false
 +CPU time usage: 0.072 seconds
 +Real time usage: 0.426 seconds
 +Preprocessor visited node count: 181/1000000
 +Preprocessor generated node count: 0/1500000
 +Post&#​8208;​expand include size: 2298/​2097152 bytes
 +Template argument size: 0/2097152 bytes
 +Highest expansion depth: 4/40
 +Expensive parser function count: 0/500
 +Unstrip recursion depth: 0/20
 +Unstrip post&#​8208;​expand size: 540/5000000 bytes
 +Number of Wikibase entities loaded: 0/400
 +Lua time usage: 0.015/​10.000 seconds
 +Lua memory usage: 1.16 MB/50 MB
 +-->
 +<!--
 +Transclusion expansion time report (%,​ms,​calls,​template)
 +100.00% ​  ​48.175 ​     1 -total
 + ​93.10% ​  ​44.849 ​     1 B&#​7843;​n_m&#​7851;​u:​Ch&​uacute;​_th&​iacute;​ch
 +  6.17%    2.970      1 B&#​7843;​n_m&#​7851;​u:​Tham_kh&#​7843;​o
 +-->
 +
 +<!-- Saved in parser cache with key viwiki:​pcache:​idhash:​55597-0!canonical!math=5 and timestamp 20181011092024 and revision id 26695253
 + ​-->​
 +</​div><​noscript><​img src="​http://​vi.wikipedia.org/​wiki/​Special:​CentralAutoLogin/​start?​type=1x1"​ alt=""​ title=""​ width="​1"​ height="​1"​ style="​border:​ none; position: absolute;"/></​noscript></​div>​
 +
 +</​HTML>​
0150--b-t-ng-th-c-h-lder-la-gi.txt · Last modified: 2018/11/07 17:09 (external edit)