Last week 1. We introduced the L p spaces: f is A -measurable L - - PowerPoint PPT Presentation

Last week

• 1. We introduced the Lp spaces:

Lp =

• f : Ω → C
• f is A-measurable
• Ω |f |p dµ < ∞
• , p ∈ [1, ∞).
• 2. And their natural seminorm

f p =

• Ω

|f |p dµ 1

p

.

• 3. We proved that the Lp spaces are vector spaces.
• 4. We proved Young’s inequality: If a, b ∈ [0, ∞) and p, q ∈ (1, ∞) are

such that 1

p + 1 q = 1, then

ab ≤ 1 p ap + 1 q bq.

Last week

• 1. We introduced the Lp spaces:

Lp =

• f : Ω → C
• f is A-measurable
• Ω |f |p dµ < ∞
• , p ∈ [1, ∞).
• 2. And their natural seminorm

f p =

• Ω

|f |p dµ 1

p

.

• 3. We proved that the Lp spaces are vector spaces.
• 4. We proved Young’s inequality: If a, b ∈ [0, ∞) and p, q ∈ (1, ∞) are

such that 1

p + 1 q = 1, then

ab ≤ 1 p ap + 1 q bq.

Last week

• 1. We introduced the Lp spaces:

Lp =

• f : Ω → C
• f is A-measurable
• Ω |f |p dµ < ∞
• , p ∈ [1, ∞).
• 2. And their natural seminorm

f p =

• Ω

|f |p dµ 1

p

.

• 3. We proved that the Lp spaces are vector spaces.
• 4. We proved Young’s inequality: If a, b ∈ [0, ∞) and p, q ∈ (1, ∞) are

such that 1

p + 1 q = 1, then

ab ≤ 1 p ap + 1 q bq.

Last week

• 1. We introduced the Lp spaces:

Lp =

• f : Ω → C
• f is A-measurable
• Ω |f |p dµ < ∞
• , p ∈ [1, ∞).
• 2. And their natural seminorm

f p =

• Ω

|f |p dµ 1

p

.

• 3. We proved that the Lp spaces are vector spaces.
• 4. We proved Young’s inequality: If a, b ∈ [0, ∞) and p, q ∈ (1, ∞) are

such that 1

p + 1 q = 1, then

ab ≤ 1 p ap + 1 q bq.

Today

• 1. We will prove H¨
• lder’s Inequality: If f , g are A-measurable and

p, q ∈ (1, ∞) are conjugate exponents ( 1

p + 1 q = 1), then

• Ω

|fg| dµ = f pgq.

• 2. We will prove Minkowski’s Inequality: If p ∈ [1, ∞) and f , g ∈ Lp,

then f + gp ≤ f p + gp.

• 3. We will discuss how to make Lp into a normed space called Lp.
• 4. We will prove that the Lp spaces are Banach spaces.
• 5. We will discuss the space of essentially bounded functions L∞.

Today

• 1. We will prove H¨
• lder’s Inequality: If f , g are A-measurable and

p, q ∈ (1, ∞) are conjugate exponents ( 1

p + 1 q = 1), then

• Ω

|fg| dµ = f pgq.

• 2. We will prove Minkowski’s Inequality: If p ∈ [1, ∞) and f , g ∈ Lp,

then f + gp ≤ f p + gp.

• 3. We will discuss how to make Lp into a normed space called Lp.
• 4. We will prove that the Lp spaces are Banach spaces.
• 5. We will discuss the space of essentially bounded functions L∞.

Today

• 1. We will prove H¨
• lder’s Inequality: If f , g are A-measurable and

p, q ∈ (1, ∞) are conjugate exponents ( 1

p + 1 q = 1), then

• Ω

|fg| dµ = f pgq.

• 2. We will prove Minkowski’s Inequality: If p ∈ [1, ∞) and f , g ∈ Lp,

then f + gp ≤ f p + gp.

• 3. We will discuss how to make Lp into a normed space called Lp.
• 4. We will prove that the Lp spaces are Banach spaces.
• 5. We will discuss the space of essentially bounded functions L∞.

Today

• 1. We will prove H¨
• lder’s Inequality: If f , g are A-measurable and

p, q ∈ (1, ∞) are conjugate exponents ( 1

p + 1 q = 1), then

• Ω

|fg| dµ = f pgq.

• 2. We will prove Minkowski’s Inequality: If p ∈ [1, ∞) and f , g ∈ Lp,

then f + gp ≤ f p + gp.

• 3. We will discuss how to make Lp into a normed space called Lp.
• 4. We will prove that the Lp spaces are Banach spaces.
• 5. We will discuss the space of essentially bounded functions L∞.

Today

• 1. We will prove H¨
• lder’s Inequality: If f , g are A-measurable and

p, q ∈ (1, ∞) are conjugate exponents ( 1

p + 1 q = 1), then

• Ω

|fg| dµ = f pgq.

• 2. We will prove Minkowski’s Inequality: If p ∈ [1, ∞) and f , g ∈ Lp,

then f + gp ≤ f p + gp.

• 3. We will discuss how to make Lp into a normed space called Lp.
• 4. We will prove that the Lp spaces are Banach spaces.
• 5. We will discuss the space of essentially bounded functions L∞.