Question

How do I prove that $\Vert f \Vert_{L^p} := \left(\int^b_a\vert f(x) \vert^p dx \right)^{\frac{1}{p}}$ defines a norm on $C^0([a,b])$?

score 90 · Answer 1 · 2026-04-04 04:40:53

90

Views

How do I prove that $\Vert f \Vert_{L^p} := \left(\int^b_a\vert f(x) \vert^p dx \right)^{\frac{1}{p}}$ defines a norm on $C^0([a,b])$?

Published on 04 Apr 2026 - 4:40

score 275 · Answer 2 · 2026-03-26 01:29:13

275

Views

Show that an inverse image $F^{-1}$ is an isometric bijection, if $F$ is an isometric bijection.

Published on 26 Mar 2026 - 1:29

score 39 · Answer 3 · 2020-01-18 21:11:33

39

Views

What's this equal: $\lim_{p\to\infty} \big(1+\frac{1}{||f||_p})^{||f||_p}?$

Published on 18 Jan 2020 - 21:11

score 82 · Answer 4 · 2026-04-08 18:45:48

82

Views

$\ker(f)$ of linear functional with operator in it.

Published on 08 Apr 2026 - 18:45

score 50 · Answer 5 · 2020-01-19 02:01:56

50

Views

About the definition of the norm $||A||$ of a linear operator $A$ in "Principles of Mathematical Analysis 3rd Edition" by Walter Rudin.

Published on 19 Jan 2020 - 2:01

score 57 · Answer 6 · 2026-04-04 04:39:20

57

Views

Defining a continuous function $f \in L^1(0,\infty), L^\infty(0,\infty) : f(t) \not\to 0$ as $t \to \infty$

Published on 04 Apr 2026 - 4:39

score 108 · Answer 7 · 2026-03-25 16:08:36

108

Views

Proving corollaries of extension of bounded linear functionals theorem

Published on 25 Mar 2026 - 16:08

score 55 · Answer 8 · 2020-01-20 14:30:58

55

Views

Closedness of the set $A = \{x \in X \mid f(x) \geq g(x)\}$

Published on 20 Jan 2020 - 14:30

score 41 · Answer 9 · 2026-04-05 23:15:40

41

Views

Show that a smooth truncating function exists on $\mathbb R^{n}$

Published on 05 Apr 2026 - 23:15

score 6.9k · Answer 10 · 2020-01-21 08:21:12

6.9k

Views

Why does $x^Tx=||x||^2$?

Published on 21 Jan 2020 - 8:21

score 168 · Answer 11 · 2020-01-21 09:30:42

168

Views

Is there any normed field such that the harmonic series converges?

Published on 21 Jan 2020 - 9:30

score 161 · Answer 12 · 2026-04-02 17:48:02

161

Views

Exact coefficient in equivalence of norm in finite dimensional space.

Published on 02 Apr 2026 - 17:48

score 28 · Answer 13 · 2020-01-21 14:38:12

28

Views

Proof of continuity of norm in function space

Published on 21 Jan 2020 - 14:38

score 44 · Answer 14 · 2026-04-01 14:54:42

44

Views

is $\|p\| = \max |p'(x)|$ norm?

Published on 01 Apr 2026 - 14:54

score 136 · Answer 15 · 2026-04-01 14:56:30

136

Views

Is $(c_{00}, \|.\|_\infty)$ not a Banach space?

Published on 01 Apr 2026 - 14:56

List Question

How do I prove that $\Vert f \Vert_{L^p} := \left(\int^b_a\vert f(x) \vert^p dx \right)^{\frac{1}{p}}$ defines a norm on $C^0([a,b])$?

Show that an inverse image $F^{-1}$ is an isometric bijection, if $F$ is an isometric bijection.

What's this equal: $\lim_{p\to\infty} \big(1+\frac{1}{||f||_p})^{||f||_p}?$

$\ker(f)$ of linear functional with operator in it.

About the definition of the norm $||A||$ of a linear operator $A$ in "Principles of Mathematical Analysis 3rd Edition" by Walter Rudin.

Defining a continuous function $f \in L^1(0,\infty), L^\infty(0,\infty) : f(t) \not\to 0$ as $t \to \infty$

Proving corollaries of extension of bounded linear functionals theorem

Closedness of the set $A = \{x \in X \mid f(x) \geq g(x)\}$

Show that a smooth truncating function exists on $\mathbb R^{n}$

Why does $x^Tx=||x||^2$?

Is there any normed field such that the harmonic series converges?

Exact coefficient in equivalence of norm in finite dimensional space.

Proof of continuity of norm in function space

is $\|p\| = \max |p'(x)|$ norm?

Is $(c_{00}, \|.\|_\infty)$ not a Banach space?

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