Function Spaces Compendium (Lp spaces) p01 Definition of Lp spaces and p-norms

1. Formula
   Observation

   On the left side of the first line of the board: $E \subset \mathbb{R}^n$ is measurable

E

a measurable set in $\mathbb{R}^n$

a subset of $\mathbb{R}^n$

1. Formula
   Observation

   On the right side of the first line of the board: f is a measurable function on E

f

a measurable function defined on the measurable set E

E

1. Formula
   Observation

   On the second line of the board: $1^\circ 0 < p < +\infty$

p

the exponent parameter in the norm definition

the open interval from zero to positive infinity

1. Formula
   Observation

   On the third line of the board: ||f||_p := ($\int_E$ |$f(x)$|^p dx)^{$\frac{1}{p}$}

||f||_p

the p-norm of the function f

the closed interval from zero to positive infinity

1. Formula
   Observation

   The fourth line on the board reads: $L^p$ := \{ $f \text{ measurable}$ : ||f||_p < +$\infty$ \}

$L^p$

the set of measurable functions whose p-norm is finite

a set of functions

1. Formula
   Observation

   The fifth line on the board reads: ||f||_\infty :=

1. the explicit mathematical expression on the right-hand side of the equals sign is not written before the clip ends

||f||_\infty

the infinity norm of function f

the closed interval from zero to positive infinity

1. Formula
   Observation

   The letter p appears multiple times on screen to distinguish the two norm definitions: $0 < p < +\infty$ and $p = +\infty$.

p

The exponent parameter of the $L^p$ space.

Zero is less than p, and p is less than or equal to positive infinity.

1. Formula
   Observation

   The screen displays $f(x)$ and |$f(x)$|, and the audio repeatedly mentions $f(x)$.

$f(x)$

A measurable function defined on the set E.

x belongs to the set E.

1. Formula
   Observation

   The screen shows $\int_E$ and $x \in E$, and the audio says 'x over E'.

E

The domain of the function f or the region of integration.

A set in a measure space.

1. Formula
   Observation

   The screen displays \|f\|_p := ($\int_E$ |$f(x)$|^p dx)^{$1/p$}.

\|f\|_p

The $L^p$ norm of the function f when $0 < p < +\infty$.

Zero is less than p, and p is less than positive infinity.

1. Formula
   Observation

   The screen displays $L^p$ := { f measurable : \|f\|_p < +$\infty$ }.

$L^p$

The space consisting of all measurable functions whose corresponding norm is finite.

Zero is less than p, and p is less than or equal to positive infinity.

1. Formula
   Observation

   The screen displays \|f\|_\infty := ess sup_{$x \in E$} |$f(x)$|.

\|f\|_\infty

The $L^\infty$ norm of the function f when $p = +\infty$.

p equals positive infinity.

1. Formula
   Observation

   The board fully writes: ||f||_p := ($\int_E$ |$f(x)$|^p dx)^{$\frac{1}{p}$}

2. Audio
   Observation

   The instructor verbally states the definition formula of this p-norm

When $0 < p < +\infty$, the p-norm of function f is defined as the one-over-p power of the integral over set E of the p-th power of the absolute value of f.

1. $E \subset \mathbb{R}^n$ is a measurable set

2. f is a measurable function on E

3. p is greater than zero and less than positive infinity

1. E
2. f
3. p

1. Formula
   Observation

   The blackboard fully writes $L^p$ := \{ $f \text{measurable}$ : ||f||_p < +$\infty$ \}

2. Audio
   Observation

   The instructor verbally states the definition of the big Lp space

The $L^p$ space is the set of all measurable functions whose p-norm is less than positive infinity.

1. The p-norm ||f||_p has been defined

1. the definition of the p-norm

1. Formula
   Observation

   At the top of the screen: 1° $0 < p < +\infty$, and \|f\|_p := ($\int_E$ |$f(x)$|^p dx)^{$1/p$}.

At the beginning of the video, the definition of the $L^p$ norm for the case $0 < p < +\infty$ is given: for a measurable function f, integrate |$f(x)$|^p over E and then raise the result to the power $1/p$. This definition serves as the reference context for the subsequent discussion of the case $p = +\infty$.

1. zero is less than p, and p is less than positive infinity

2. f is a measurable function

3. the domain of integration is E

1. Formula
   Observation

   the screen displays $L^p$ := { f measurable : \|f\|_p < +$\infty$ }.

the video defines $L^p$ as the set of all measurable functions for which the corresponding norm is finite. The clip first shows the notation $0 < p < +\infty$, then proceeds to define the norm when $p = +\infty$.

1. f is measurable

2. the p-norm of f is less than positive infinity

1. definition of the $L^p$ norm for finite exponent p
2. definition of the $L^\infty$ norm when $p = +\infty$

1. Formula
   Observation

   the screen displays 2° $p = +\infty$, and \|f\|_\infty := ess sup_{$x \in E$} |$f(x)$|.

2. Audio
   Observation

   the audio says, "this symbol denotes the essential supremum of the absolute value of $f(x)$ over x in E."

the video explicitly states that when $p = +\infty$, the integral-type formula is no longer used; instead, \|f\|_\infty is defined as the essential supremum of |$f(x)$| over E. It then further expands the essential supremum as the infimum of the set of essential upper bounds.

1. p equals positive infinity

2. f is a measurable function

3. Consider $x \in E$

1. Definition of essential supremum

1. Audio
   Observation

   The audio says, "How do we define the essential supremum? We define it as the smallest among all essential upper bounds—the smallest one is naturally the infimum."

2. Formula
   Observation

   The screen displays ess sup_{$x \in E$} |$f(x)$| = inf { M : |$f(x)$| $\leqslant M$, a.e. $x \in E$ }.

The video explains ess sup as "the smallest among all essential upper bounds" and further clarifies that "the smallest one" is the infimum of the set of all such essential upper bounds. Thus, the essential supremum is not the ordinary supremum, but the infimum of the set of upper bounds that hold almost everywhere.

1. f is measurable on E

2. M ranges over all real numbers such that |$f(x)$| $\leqslant M$ holds almost everywhere

1. Definitions of essential upper bound and essential boundedness

1. Formula
   Observation

   The screen writes: "If there exists $M > 0$ such that |$f(x)$| $\leqslant M$ for almost every $x \in E$, then $f(x)$ is said to be essentially bounded on E, and M is called an essential upper bound."

2. Audio
   Observation

   The audio says, "If it holds for every x, then M is an upper bound for |$f(x)$|. However, what does essential upper bound mean here? We do not require it to hold at every point—everywhere—but only almost everywhere."

The video first recalls the ordinary upper bound: if there exists $M > 0$ such that |$f(x)$| $\leqslant M$ holds for all x, then M is an upper bound and f is bounded. It then emphasizes that an essential upper bound replaces 'holds everywhere' with 'holds almost everywhere', meaning it may fail on a set of measure zero; a function f satisfying this condition is said to be essentially bounded on E, and the corresponding M is called an essential upper bound.

1. M is greater than zero

2. |$f(x)$| $\leqslant M$ holds almost everywhere on E

1. Formula
   Observation

   The screen gradually writes $L^{\infty}:=\{f\text{measurable}:\ \|f\|_{\infty}<+\infty\}$.

2. Audio
   Observation

   The instructor says, "The space of essentially bounded $L$ is defined as follows: first, $f$ must be measurable; additionally, the infinity norm of $f$ must be finite."

This segment defines $L^{\infty}$ as the set of all measurable functions whose infinity norms are finite. Its definition matches the previously given $L^p$ in form, except that the norm condition is changed from $\|f\|_p<+\infty$ to $\|f\|_{\infty}<+\infty$.

1. $f$ is measurable

2. The infinity norm of f is less than positive infinity

1. The infinity norm equals the essential supremum

1. Formula
   Observation

   The screen retains $\|f\|_{\infty}:=\operatorname*{ess\,sup}_{x\in E}|f(x)|=\inf\{M:\ |f|\le M\ \text{a.e.}\ x\in E\}$.

2. Audio
   Observation

   The instructor concludes, "The infinity norm of $f$ is the essential supremum."

The video defines the infinity norm of a function as $|f(x)|$ on $E$ the essential supremum, and further writes it as the infimum of those $|f|$ such that $M$ is almost everywhere less than or equal to $M$ . This shows that the infinity norm is not the ordinary pointwise supremum, but rather the smallest essential bound after ignoring sets of measure zero.

1. $f$ is a measurable function on $E$

2. Involves upper bounds in the almost-everywhere sense

1. Formula
   Observation

   The screen displays: "If $\exists M>0$, such that $|f(x)|\le M$ almost everywhere on $x\in E$, then $f(x)$ is said to be essentially bounded on $E$, and $M$ is called an essential upper bound."

The video presents the criterion for "essential boundedness": if there exists a positive number $M$ such that $|f(x)|\le M$ holds almost everywhere on $E$, then $f$ is said to be essentially bounded on $E$, and such an $M$ is called an essential upper bound. This concept directly supports the meaning of $\|f\|_{\infty}$ as the greatest lower bound.

1. M is greater than zero

2. The inequality needs only to hold almost everywhere

1. Audio
   Observation

   The instructor concludes: "Thus we have defined the $f$ norm of $p$, the L-$L^p$ space, and the infinity norm of $f$, as well as the L-infinity space of $L$."

2. Formula
   Observation

   The screen simultaneously retains the four definitions: $\|f\|_p$, $L^p$, $\|f\|_{\infty}$, and $L^{\infty}$.

At the end of the segment, the four previously written objects are reviewed collectively: $\|f\|_p$ is defined via integration; $L^p$ is the space of measurable functions for which it is finite; $\|f\|_{\infty}$ is defined via the essential supremum; and $L^{\infty}$ is the space of measurable functions for which it is finite. Here, the term "method" refers to an abstraction of the isomorphic definition pattern.

1. applies to the $0<p<+\infty$ case of $L^p$ ,

2. applies to the $p=+\infty$ case of $L^{\infty}$ .

1. Definition of the $L^{\infty}$ space
2. The infinity norm equals the essential supremum

1. Audio
   Observation

   The instructor says, "I want to define the infinity norm of f; actually, its definition is the essential supremum."

2. Formula
   Observation

   The blackboard writes ||f||_\infty := but the right-hand side remains incomplete

1. The precise mathematical expression for the essential supremum is not given in this segment

The infinity norm ||f||_\infty of the function f is defined as its essential supremum.

1. f is a measurable function on E

For the measurable function f

1. Audio
   Observation

   The audio explains step by step: first define what an essential upper bound is, then define the essential supremum as the smallest among all essential upper bounds, and the smallest such value is precisely the infimum.

2. Formula
   Observation

   The screen finally displays ess sup_{$x \in E$} |$f(x)$| = inf { M : |$f(x)$| $\leqslant M$, a.e. $x \in E$ }.

1. Expression

   $$\exists M > 0, \text{ s.t. } |f(x)| \leqslant M, \text{ a.e. } x \in E$$
   Explanation

   First identify which values M can serve as essential upper bounds for |$f(x)$|, i.e., the inequality needs only to hold almost everywhere.

   Justification

   The spoken definition of 'essential upper bound' in the audio matches the written definition on the board.

   Shown in the video
2. Expression

   $$\{ M : |f(x)| \leqslant M, \text{a.e. } x \in E \}$$
   Explanation

   Collect all such M satisfying the above condition into a set.

   Justification

   This set is prepared as the object for taking 'the smallest one'.

   Shown in the video
3. Expression

   $$ess sup_{x \in E} |f(x)| = \inf \{ M : |f(x)| \leqslant M, \text{a.e. } x \in E \}$$
   Explanation

   Then take the infimum of this set to obtain the essential supremum.

   Justification

   The audio explicitly states: 'The essential supremum is the smallest among all essential upper bounds; the smallest one is, of course, the infimum.'

   Shown in the video

The essential supremum can be written as "the infimum of all M such that |$f(x)$| $\leqslant M$ holds almost everywhere".

1. Audio
   Observation

   The instructor repeats in sequence $f$ of $p$ the norm is the integral of the absolute value to the power of $p$ then raised to the power of one over $p$ one over$f$ the infinity norm is the essential supremum $L^p$ the space and $L$ the infinity space are actually the spaces formed by those functions for which the corresponding norm is finite.

2. Formula
   Observation

   The screen simultaneously displays the full definitions of $\|f\|_p$, $L^p$, $\|f\|_{\infty}$, and $L^{\infty}$.

1. Expression

   $$$$
   Explanation

   First, interpret $\|f\|_p$ as "the integral of $|f|^p$ raised to the $1/p$th power".

   Justification

   Corresponds to the formula $\|f\|_p:=\left(\int_E |f(x)|^p dx\right)^{1/p}$ already shown at the top of the screen.

   Shown in the video
2. Expression

   $$$$
   Explanation

   Next, interpret $\|f\|_{\infty}$ as "essential supremum".

   Justification

   Corresponds to $\|f\|_{\infty}:=\operatorname*{ess\,sup}_{x\in E}|f(x)|$ in the middle of the screen.

   Shown in the video
3. Expression

   $$$$
   Explanation

   Finally, clarify that both $L^p$ and $L^{\infty}$ denote the collection of all measurable functions whose corresponding norms are finite.

   Justification

   Corresponds to the set definitions $L^p:=\{f\text{measurable}:\ \|f\|_p<+\infty\}$ and $L^{\infty}:=\{f\text{measurable}:\ \|f\|_{\infty}<+\infty\}$ shown on screen.

   Shown in the video

This segment orally recaps the four definitions—$p$ norm, $L^p$, infinity norm, and $L^{\infty}$—as a unified construction: first define a norm, then take the collection of all measurable functions with finite norm.

1. Animation
   Observation

   The symbols ||f||_p, $L^p$, and ||f||_\infty appear sequentially in handwriting on screen, along with their respective definition formulas

1. Handwritten script

2. Grid background

1. Writing begins at the upper right corner with ||f||_p and $L^p$

2. Then, $1^\circ 0 < p < +\infty$ and its specific definition formula are written on the left

3. Finally, $2^\circ p = +\infty$ and ||f||_\infty := are written

1. The grid background remains unchanged

2. The top title $L^p$ space remains unchanged

The visual presentation shows the logical derivation and board layout from the general p-norm to the special case $p = +\infty$.

1. Animation
   Observation

   The screen shows a digital whiteboard with handwritten steps: first displaying the two existing parts $0 < p < +\infty$ and $p = +\infty$, then gradually completing the set definition after ess sup, and writing the Chinese explanation of essential upper bound below.

1. Formula \|f\|_p

2. Set definition of $L^p$

3. Formula \|f\|_\infty

4. Text "essential"

5. Formula inf { M : |$f(x)$| $\leqslant M$, a.e. $x \in E$ }

6. Chinese definition sentence: "If there exists $M > 0$ ... then $f(x)$ is said to be essentially bounded on E, and M is called an essential upper bound."

1. From second 0 to second 12: the screen already displays the main part of \|f\|_\infty := ess sup_{$x \in E$} |$f(x)$| for $p = +\infty$.

2. From second 12 to second 34: the presenter gives a verbal explanation of "essential supremum" and writes the characters "essential" beside it.

3. From second 34 to second 46: the expression "= inf { M : " is added after ess sup.

4. From 46 to 86 seconds: On a new line below, write "If ∃ $M > 0$ such that |$f(x)$| ≤ M.".

5. From 86 to 120 seconds: Continue completing "a.e. $x \in E$", and write "then $f(x)$ is said to be essentially bounded on E, and M is called an essential upper bound.".

6. From 120 to 144 seconds: Return to the formula above and complete the condition inside the inf set as |$f(x)$| ≤ M, a.e. $x \in E$.

1. Always focus on the definition of the $L^p$ space and the norm for $p = +\infty$.

2. The core object has consistently been the upper-bound property of |$f(x)$| on E.

The visual process demonstrates the blackboard progression from "first introducing the symbol ess sup" to "then explaining its set-theoretic definition", helping ground the abstract notation in "the infimum of the set of essential upper bounds".

1. Animation
   Observation

   At the bottom of the screen, $L^{\infty}$, $:=\{$, "measurable" for $f$, and $:\ \|f\|_{\infty}<+\infty\}$ appear sequentially from blank space.

2. Audio
   Observation

   While explaining, the speaker says: "The space big $L$ infinity is defined... First, $f$ must be measurable; the additional requirement is that the infinity norm of $f$ must be finite."

1. the space of essentially bounded integrable functions

2. $f$ is measurable

3. the infinity norm of f is finite

1. first write the space notation $L^{\infty}$

2. then write the set notation and the element condition "$f$ is measurable"

3. finally add the finite-norm condition $\|f\|_{\infty}<+\infty$

1. the existing definitions of $\|f\|_p$, $L^p$, and $\|f\|_{\infty}$ above remain visible on screen

2. the writing toolbar and page layout remain unchanged

visually demonstrates that the definition of $L^{\infty}$ is not given in isolation, but follows the set-theoretic notation for $L^p$ above, replacing only the norm condition with finiteness of the infinity norm

1. Animation
   Observation

   approximately 23 seconds later, the view zooms out to fully reveal the title "$L^p$ space", the assumptions "$E\subset \mathbb{R}^n$ is measurable, $f$ is a measurable function on $E$", and the full content of both parts $1^\circ$ and $2^\circ$

1. title "$L^p$ space"

2. The premise line reads "$E\subset \mathbb{R}^n$ is measurable, and $f$ is a measurable function on $E$."

3. the $1^\circ\ 0<p<+\infty$ part

4. the $2^\circ\ p=+\infty$ part

1. The screen switches from the local writing area to the full-page overview

2. The originally separate upper and lower definitions now appear together in view.

1. The textual content itself is unchanged—only the display range has expanded.

This zoom reintegrates the newly written $L^{\infty}$ into the full-page layout, making it clear that the definitions of $p<\infty$ and $p=\infty$ unfold side by side.

1. Audio
   Observation

   The instructor says, 'In fact, at the very beginning of the definition, we certainly have not yet verified that it is a norm; however, we formally define it this way first, and then gradually verify that it indeed satisfies all the axioms of a norm.'

The misconception that writing down the formula for ||f||_p instantly proves it satisfies all the norm axioms.

This is merely a formal symbolic definition; later, we must verify step by step that it truly satisfies the properties of a norm.

1. Audio
   Observation

   The instructor says, 'We first call it a space; in fact, it is initially just a set—we verify step by step that it is a linear space... and ultimately a Banach space.'

The misconception that $L^p$ inherently possesses the full Banach space structure from the outset.

$L^p$ begins simply as the set of functions satisfying certain conditions; its linear space structure, normed linear space structure, and completeness (i.e., Banach space structure) must be verified step by step.

1. Audio
   Observation

   The audio emphasizes "I do not demand the inequality to hold at every point—i.e., everywhere—but only almost everywhere.".

It is easy to mistakenly interpret "essential upper bound" as requiring |$f(x)$| ≤ M to hold at every point of E.

The video explicitly states that an essential upper bound only requires the inequality to hold almost everywhere, allowing failure on a set of measure zero; this is precisely what the term "essential" means.

1. Formula
   Observation

   The screen explicitly writes $\operatorname*{ess\,sup}$, with the condition marked "a.e.", rather than the ordinary $\sup$.

1. This is a caution raised by the analyst based on the on-screen formula; the video does not verbally emphasize this pitfall.

Upon seeing $\|f\|_{\infty}$, one might mistakenly interpret it as the supremum of pointwise maxima.

The definition given in the video is $\operatorname*{ess\,sup}_{x\in E}|f(x)|=\inf\{M:\ |f|\le M\ \text{a.e.}\ x\in E\}$; the key lies in the "almost everywhere" restriction, which permits exceptions on a set of measure zero—thus it is essentially the smallest essential upper bound, not an ordinary pointwise supremum.

1. Formula
   Observation

   The definition formula of $L^p$ directly uses the symbol ||f||_p

The definition of the $L^p$ space depends on the definition of the p-norm: only after defining ||f||_p can we use ||f||_p < +$\infty$ to select the elements belonging to $L^p$.

1. Formula
   Observation

   In the visual, \|f\|_\infty is directly defined as ess sup_{$x \in E$} |$f(x)$|.

The definition of the $L^\infty$ norm directly uses the concept of essential supremum.

1. Audio
   Observation

   The audio first says, "Next, we define what an essential upper bound is," then says, "The essential supremum is the smallest among all essential upper bounds."

To understand the essential supremum, one must first understand what an essential upper bound is; the essential supremum is the infimum of the set of all essential upper bounds.

1. Formula
   Observation

   The screen displays two parallel definitions of norms: 1° $0 < p < +\infty$ and 2° $p = +\infty$.

The norm for finite p is defined via integration, whereas the norm for $p = +\infty$ is defined via the essential supremum, forming a side-by-side comparison.

1. Formula
   Observation

   On screen: $L^p$ := { f measurable : \|f\|_p < +$\infty$ }, and \|f\|_p for $p = +\infty$ is again defined using ess sup.

The definition of the $L^p$ space depends on whether the corresponding norm is finite; when $p = +\infty$, this condition becomes whether the essential supremum is finite.

1. Formula
   Observation

   $L^{\infty}:=\{f\text{measurable}:\ \|f\|_{\infty}<+\infty\}$ directly quotes the definition of $\|f\|_{\infty}$ from the previous line.

To understand why $L^{\infty}$ is defined as 'the set of all measurable functions with finite infinity norm', one must first grasp that the infinity norm is essentially the essential supremum.

1. Formula
   Observation

   The screen displays $L^p:=\{f\text{measurable}:\ \|f\|_p<+\infty\}$ and $L^{\infty}:=\{f\text{measurable}:\ \|f\|_{\infty}<+\infty\}$ side by side.

2. Audio
   Observation

   While summarizing, the instructor presents the 'big $L^p$ space' and the '$L$ infinity space' together.

Both are constructed using the same set-forming principle—'measurable and having finite corresponding norm'—but employ different norms: one is the integral-type $p$ norm, the other is the essential-supremum-type infinity norm.

1. Formula
   Observation

   The definition of 'essentially bounded' appears alongside $\inf\{M:\ |f|\le M\ \text{a.e.}\ x\in E\}$ in the same section.

The infinity norm is expressed as the infimum of certain essential upper bounds $M$, so the concepts of 'essentially bounded' and 'essential upper bound' form the immediate foundation for understanding the expression $\|f\|_{\infty}$.

1. Formula
   Observation

   The blackboard displays the complete integral formula for the p-norm

1. the definition of the p-norm

1. Formula
   Observation

   The blackboard presents the descriptive definition of the $L^p$ set.

1. Definition of the $L^p$ space

1. Audio
   Observation

   The instructor verbally defines the infinity norm as the essential supremum.

1. The infinity norm equals the essential supremum

1. Audio
   Observation

   The audio directly asks and answers, "How is the essential supremum defined?"

1. Definition of essential supremum
2. Definitions of essential upper bound and essential boundedness

1. Formula
   Observation

   The screen writes \|f\|_\infty := ess sup_{$x \in E$} |$f(x)$| for $p = +\infty$.

1. definition of the $L^\infty$ norm when $p = +\infty$
2. Definition of essential supremum

1. Audio
   Observation

   The audio contrasts "holding everywhere" with "holding almost everywhere."

1. Definitions of essential upper bound and essential boundedness

1. Audio
   Observation

   The audio says, "Because the fact that it holds almost everywhere reveals some essential properties of this function, it is called an essential upper bound."

1. Definitions of essential upper bound and essential boundedness
2. Do not confuse an essential upper bound with an ordinary pointwise upper bound.

1. Formula
   Observation

   The screen is writing out $L^{\infty}:=\{f\text{measurable}:\ \|f\|_{\infty}<+\infty\}$.

1. Definition of the $L^{\infty}$ space

1. Formula
   Observation

   The screen displays $\|f\|_{\infty}:=\operatorname*{ess\,sup}_{x\in E}|f(x)|=\inf\{M:\ |f|\le M\ \text{a.e.}\ x\in E\}$.

1. The infinity norm equals the essential supremum
2. Do not confuse the essential supremum with the ordinary supremum.

1. Formula
   Observation

   The screen states: 'If $\exists M>0$, such that $|f(x)|\le M$ almost everywhere $x\in E$, then $f(x)$ is said to be essentially bounded on $E$.'

1. Essential boundedness and essential upper bounds

1. Audio
   Observation

   At the end, the instructor summarizes that both $L^p$ and $L^{\infty}$ are function spaces where the corresponding norm is finite.

1. Summary of the $p$ norm, $L^p$, the infinity norm, and $L^{\infty}$
2. s288-cr-Lp-Linfty-parallel