Question

How $\|f\|=\sup\left\{\frac{\vert f(x)\vert}{\|x\|}:x \ne 0,x\in X \right\}$ and $\|f\|=\sup\{\vert f(x)\vert:\|x\|=1 ,x\in X \}$ are equivalent?

score 73 · Answer 1 · 2026-04-14 21:09:02

73

Views

How $\|f\|=\sup\left\{\frac{\vert f(x)\vert}{\|x\|}:x \ne 0,x\in X \right\}$ and $\|f\|=\sup\{\vert f(x)\vert:\|x\|=1 ,x\in X \}$ are equivalent?

Published on 14 Apr 2026 - 21:09

score 52 · Answer 2 · 2026-04-12 14:54:55

52

Views

How $S_1$ & $S_2$ are equivalent?

Published on 12 Apr 2026 - 14:54

score 66 · Answer 3 · 2026-04-15 10:33:45

66

Views

Proving that the operator $G: L_p[0,1] \to L_q[0,1]; G(x)=x$ is linear, well defined and continuous. And find it's norm.

Published on 15 Apr 2026 - 10:33

score 527 · Answer 4 · 2026-04-12 18:21:20

527

Views

How do I show that the limit of the quotient of $n^{th}$ and $(n+1)^{th}$ norm of a function equals its infinity norm?

Published on 12 Apr 2026 - 18:21

score 75 · Answer 5 · 2026-04-16 02:27:56

75

Views

If $f\in L^2([0,1]^2)$, what can we say about $g(x)=f(x,x)$?

Published on 16 Apr 2026 - 2:27

score 359 · Answer 6 · 2026-04-08 19:57:00

359

Views

How to prove the following theorem-

Published on 08 Apr 2026 - 19:57

score 35 · Answer 7 · 2026-04-13 12:41:43

35

Views

Need help in proving a small part of a theorem-

Published on 13 Apr 2026 - 12:41

score 558 · Answer 8 · 2026-04-15 12:42:53

558

Views

Hölder inequality for n functions

Published on 15 Apr 2026 - 12:42

score 250 · Answer 9 · 2026-04-13 17:10:25

Alternative definition of $\|f\|_{\infty}$ as the smallest of all numbers of the form $\sup\{|g(x)| : x \in X \}$, where $f = g$ almost everywhere

score 67 · Answer 10 · 2026-04-11 19:07:22

67

Views

Proof Check (Hardy Maximal Function )

Published on 11 Apr 2026 - 19:07

score 37 · Answer 11 · 2026-04-15 21:24:39

37

Views

When we eradicate one element from a complete space, how to reach the completeness of this new space?

Published on 15 Apr 2026 - 21:24

score 296 · Answer 12 · 2026-04-17 05:08:06

296

Views

Isometry between two $L^p$ spaces

Published on 17 Apr 2026 - 5:08

score 40 · Answer 13 · 2026-04-15 09:52:52

40

Views

Embedding of a closed subspace of $L^1_m([0,1])$ with each element in some $L^p$ with $p>1$ in a higher $L^{p_0}$.

Published on 15 Apr 2026 - 9:52

score 231 · Answer 14 · 2026-04-15 02:10:28

231

Views

How to show that $L_p(\mu)_+$ is not solid (i.e., $L_p(\mu)_+$ has empty interior)?

Published on 15 Apr 2026 - 2:10

score 59 · Answer 15 · 2026-04-16 04:32:34

59

Views

Assume that $f,g \in L_{10}(\mu)$. Prove that their product $fg$ belongs to $L_5(\mu)$.

Published on 16 Apr 2026 - 4:32

List Question

How $\|f\|=\sup\left\{\frac{\vert f(x)\vert}{\|x\|}:x \ne 0,x\in X \right\}$ and $\|f\|=\sup\{\vert f(x)\vert:\|x\|=1 ,x\in X \}$ are equivalent?

How $S_1$ & $S_2$ are equivalent?

Proving that the operator $G: L_p[0,1] \to L_q[0,1]; G(x)=x$ is linear, well defined and continuous. And find it's norm.

How do I show that the limit of the quotient of $n^{th}$ and $(n+1)^{th}$ norm of a function equals its infinity norm?

If $f\in L^2([0,1]^2)$, what can we say about $g(x)=f(x,x)$?

How to prove the following theorem-

Need help in proving a small part of a theorem-

Hölder inequality for n functions

Alternative definition of $\|f\|_{\infty}$ as the smallest of all numbers of the form $\sup\{|g(x)| : x \in X \}$, where $f = g$ almost everywhere

Proof Check (Hardy Maximal Function )

When we eradicate one element from a complete space, how to reach the completeness of this new space?

Isometry between two $L^p$ spaces

Embedding of a closed subspace of $L^1_m([0,1])$ with each element in some $L^p$ with $p>1$ in a higher $L^{p_0}$.

How to show that $L_p(\mu)_+$ is not solid (i.e., $L_p(\mu)_+$ has empty interior)?

Assume that $f,g \in L_{10}(\mu)$. Prove that their product $fg$ belongs to $L_5(\mu)$.

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