Question

Prove that $\Vert f(b)-f(a)-f'(a)(b-a) \Vert\leq \sup_{x\in [a,b]} \Vert f''(x)\Vert\Vert b-a\Vert^2$

score 187 · Answer 1 · 2018-08-17 16:18:34

187

Views

Prove that $\Vert f(b)-f(a)-f'(a)(b-a) \Vert\leq \sup_{x\in [a,b]} \Vert f''(x)\Vert\Vert b-a\Vert^2$

Published on 17 Aug 2018 - 16:18

score 113 · Answer 2 · 2026-03-25 16:44:23

113

Views

Any two norms on a finite dimensional $\Bbb K$-linear space are equivalent.

Published on 25 Mar 2026 - 16:44

score 955 · Answer 3 · 2018-08-19 07:24:32

955

Views

Are finite dimensional normed linear spaces locally compact?

Published on 19 Aug 2018 - 7:24

score 1.2k · Answer 4 · 2026-03-26 07:35:40

1.2k

Views

Fourier Sine series uses the orthogonal set $\{\sin(nx)\}^\infty_{n = 1}$ on $0 \le x \le \pi$

Published on 26 Mar 2026 - 7:35

score 1.3k · Answer 5 · 2018-08-21 08:46:53

1.3k

Views

Finite-dimensional subspaces of normed linear spaces are closed: Do there exist alternative proofs?

Published on 21 Aug 2018 - 8:46

score 11.1k · Answer 6 · 2018-08-21 15:28:28

11.1k

Views

All norms of $\mathbb R^n$ are equivalent

Published on 21 Aug 2018 - 15:28

score 75 · Answer 7 · 2018-08-22 01:07:24

75

Views

Easy way to see that $|x-z|^2 \geq |y-z|^2 + (2y-2z)\cdot (x-y),\forall x,y \in \mathbb{R}^n$, for fixed $z \in \mathbb{R}^n$

Published on 22 Aug 2018 - 1:07

score 74 · Answer 8 · 2026-03-28 15:18:53

74

Views

Question regarding $l_2$-space.

Published on 28 Mar 2026 - 15:18

score 95 · Answer 9 · 2026-03-25 04:43:02

95

Views

$f'$ is bounded and $\sup\limits_{c\in O}\Vert f'(c) \Vert<\infty$ if and only if $f$ is globally Lipschitz

Published on 25 Mar 2026 - 4:43

score 809 · Answer 10 · 2026-03-29 04:41:26

809

Views

What is the dimension of $l_p$-space, $1 \leq p < \infty$?

Published on 29 Mar 2026 - 4:41

score 206 · Answer 11 · 2018-08-23 05:17:38

206

Views

Prove that $\langle f'(x)h,h\rangle\geq (1-k)\ \Vert h\Vert^2$ and that $\lim\limits_{\Vert x\Vert\to \infty}f(x)=\infty$

Published on 23 Aug 2018 - 5:17

score 741 · Answer 12 · 2018-08-23 07:14:43

741

Views

Operator Norm from Geometrical Point of View

Published on 23 Aug 2018 - 7:14

score 108 · Answer 13 · 2018-08-23 09:40:51

108

Views

If a subspace $V_1$ is close to a subspace $V_2$ of the same dimension, is $V_2$ close to $V_1$?

Published on 23 Aug 2018 - 9:40

score 460 · Answer 14 · 2026-03-25 03:06:59

460

Views

Does the modulus of a linear operator change continuously with the operator?

Published on 25 Mar 2026 - 3:06

score 40 · Answer 15 · 2026-03-25 06:29:35

40

Views

Showing $ \left\lVert P_C(x) - x \right\rVert_2^2 \le \left\lVert x - c \right\rVert_2^2 - \left\lVert P_C(x) - c \right\rVert_2^2 $

Published on 25 Mar 2026 - 6:29

List Question

Prove that $\Vert f(b)-f(a)-f'(a)(b-a) \Vert\leq \sup_{x\in [a,b]} \Vert f''(x)\Vert\Vert b-a\Vert^2$

Any two norms on a finite dimensional $\Bbb K$-linear space are equivalent.

Are finite dimensional normed linear spaces locally compact?

Fourier Sine series uses the orthogonal set $\{\sin(nx)\}^\infty_{n = 1}$ on $0 \le x \le \pi$

Finite-dimensional subspaces of normed linear spaces are closed: Do there exist alternative proofs?

All norms of $\mathbb R^n$ are equivalent

Easy way to see that $|x-z|^2 \geq |y-z|^2 + (2y-2z)\cdot (x-y),\forall x,y \in \mathbb{R}^n$, for fixed $z \in \mathbb{R}^n$

Question regarding $l_2$-space.

$f'$ is bounded and $\sup\limits_{c\in O}\Vert f'(c) \Vert<\infty$ if and only if $f$ is globally Lipschitz

What is the dimension of $l_p$-space, $1 \leq p < \infty$?

Prove that $\langle f'(x)h,h\rangle\geq (1-k)\ \Vert h\Vert^2$ and that $\lim\limits_{\Vert x\Vert\to \infty}f(x)=\infty$

Operator Norm from Geometrical Point of View

If a subspace $V_1$ is close to a subspace $V_2$ of the same dimension, is $V_2$ close to $V_1$?

Does the modulus of a linear operator change continuously with the operator?

Showing $ \left\lVert P_C(x) - x \right\rVert_2^2 \le \left\lVert x - c \right\rVert_2^2 - \left\lVert P_C(x) - c \right\rVert_2^2 $

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