Analysis
21 videos • 435 views • by Math-Pi
1
Group and Abelian group: Explaining with expanding the examples.
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2
Vector Norms: Explained with examples and the way to sketch their graphs.
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3
Matrix Norms: L-1, L-2, L- ∞, and Frobenius norm explained with examples.
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4
Dominated convergence theorem: Explained with an examples.
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5
Monotone Convergence Theorem: Explained with graph and examples.
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6
First Fundamental Theorem of calculus: Statement and Proof.
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7
2nd Fundamental Theorem of Calculus: Statement and Proof.
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8
Weierstrass M-test: statement and examples
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9
Mean Value Theorem (for integrals): Statement and proof.
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10
Darboux Theorem: Statement and proof.
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11
Intermediate Value Theorem: Statement and proof.
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12
Rolle's Theorem: Statement and proof.
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13
Product rule
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14
Quotient Rule for Derivatives: Statement and proof.
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15
Theorem: Every differentiable function is continuous but the converse is not true.
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16
Filter: Deinition and some examples.
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17
Theorem: Similar matrices have the same characteristic-polynomials/eigenvalues/trace/determinant.
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18
Prove that every Hilbert space H has an orthonormal basis (use the Zorn’s lemma).
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19
Definition and Example: Increasing, Decreasing and Monotonic sequence.
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20
Thm (uniform convergence): The uniform limit commute with the Riemann integral.
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21
Hilbert Space vs. Banach Space : Trick to distinguish between these two vector spaces.
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