Continuous mappings | Mathematical analysis
In mathematical analysis, a family of functions is equicontinuous if all the functions are continuous and they have equal variation over a given neighbourhood, in a precise sense described herein. In particular, the concept applies to countable families, and thus sequences of functions. Equicontinuity appears in the formulation of Ascoli's theorem, which states that a subset of C(X), the space of continuous functions on a compact Hausdorff space X, is compact if and only if it is closed, pointwise bounded and equicontinuous. As a corollary, a sequence in C(X) is uniformly convergent if and only if it is equicontinuous and converges pointwise to a function (not necessarily continuous a-priori). In particular, the limit of an equicontinuous pointwise convergent sequence of continuous functions fn on either metric space or locally compact space is continuous. If, in addition, fn are holomorphic, then the limit is also holomorphic. The uniform boundedness principle states that a pointwise bounded family of continuous linear operators between Banach spaces is equicontinuous. (Wikipedia).
The method of determining eigenvalues as part of calculating the sets of solutions to a linear system of ordinary first-order differential equations.
From playlist A Second Course in Differential Equations
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From playlist LINEAR ALGEBRA 3: EIGENVALUES AND EIGENVECTORS
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From playlist A Second Course in Differential Equations
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From playlist Geometry
Metric Spaces - Lectures 21, 22 & 23: Oxford Mathematics 2nd Year Student Lecture
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Review of previous results. Equicontinuity. Exercise: finite set of uniformly continuous functions is equicontinuous. A uniformly convergent sequence of continuous functions on a compact set is equicontinuous. Theorem of Ascoli-Arzela: a pointwise bounded sequence of equicontinuous fun
From playlist Course 7: (Rudin's) Principles of Mathematical Analysis (Fall 2018)
Functional Analysis - Part 17 - Arzelà–Ascoli theorem
Support the channel on Steady: https://steadyhq.com/en/brightsideofmaths Or support me via PayPal: https://paypal.me/brightmaths Functional analysis series: https://www.youtube.com/playlist?list=PLBh2i93oe2qsGKDOsuVVw-OCAfprrnGfr PDF versions: https://steadyhq.com/en/brightsideofmaths/po
From playlist Functional analysis
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From playlist GEOMETRY CH 1 BASIC CONCEPTS
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Changing notation with complex eigenvalues.
From playlist A Second Course in Differential Equations
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From playlist Course 7: (Rudin's) Principles of Mathematical Analysis (Fall 2018)