Last week 1. We talked about some Hilbert space facts you knew from - - PowerPoint PPT Presentation

Last week

• 1. We talked about some Hilbert space facts you knew from before.
• 2. We looked at the concept of “nearest point in a subset” and the
• rthogonal projection operators.
• 3. We introduced the Lp spaces:

Lp =

• f : Ω → C
• f is A-measurable
• Ω |f |p dµ < ∞
• , p ∈ (0, ∞).
• 4. And their natural seminorm

f p =

• Ω

|f |p dµ 1

p

.

• 5. We proved that the Lp spaces are linear spaces.
• 6. We proved H¨
• lder’s Inequality.

Last week

• 1. We talked about some Hilbert space facts you knew from before.
• 2. We looked at the concept of “nearest point in a subset” and the
• rthogonal projection operators.
• 3. We introduced the Lp spaces:

Lp =

• f : Ω → C
• f is A-measurable
• Ω |f |p dµ < ∞
• , p ∈ (0, ∞).
• 4. And their natural seminorm

f p =

• Ω

|f |p dµ 1

p

.

• 5. We proved that the Lp spaces are linear spaces.
• 6. We proved H¨
• lder’s Inequality.

Last week

• 1. We talked about some Hilbert space facts you knew from before.
• 2. We looked at the concept of “nearest point in a subset” and the
• rthogonal projection operators.
• 3. We introduced the Lp spaces:

Lp =

• f : Ω → C
• f is A-measurable
• Ω |f |p dµ < ∞
• , p ∈ (0, ∞).
• 4. And their natural seminorm

f p =

• Ω

|f |p dµ 1

p

.

• 5. We proved that the Lp spaces are linear spaces.
• 6. We proved H¨
• lder’s Inequality.

Last week

• 1. We talked about some Hilbert space facts you knew from before.
• 2. We looked at the concept of “nearest point in a subset” and the
• rthogonal projection operators.
• 3. We introduced the Lp spaces:

Lp =

• f : Ω → C
• f is A-measurable
• Ω |f |p dµ < ∞
• , p ∈ (0, ∞).
• 4. And their natural seminorm

f p =

• Ω

|f |p dµ 1

p

.

• 5. We proved that the Lp spaces are linear spaces.
• 6. We proved H¨
• lder’s Inequality.

Last week

• 1. We talked about some Hilbert space facts you knew from before.
• 2. We looked at the concept of “nearest point in a subset” and the
• rthogonal projection operators.
• 3. We introduced the Lp spaces:

Lp =

• f : Ω → C
• f is A-measurable
• Ω |f |p dµ < ∞
• , p ∈ (0, ∞).
• 4. And their natural seminorm

f p =

• Ω

|f |p dµ 1

p

.

• 5. We proved that the Lp spaces are linear spaces.
• 6. We proved H¨
• lder’s Inequality.

Today

• 1. We will prove Minkowski’s Inequality:

f + gp ≤ f p + gp.

• 2. We will discuss how to make Lp into a normed space called Lp.
• 3. We will prove that the Lp spaces are Banach spaces.
• 4. In particular L2 is a Hilbert space with inner product given by

f , g =

• Ω

f g dµ.

Today

• 1. We will prove Minkowski’s Inequality:

f + gp ≤ f p + gp.

• 2. We will discuss how to make Lp into a normed space called Lp.
• 3. We will prove that the Lp spaces are Banach spaces.
• 4. In particular L2 is a Hilbert space with inner product given by

f , g =

• Ω

f g dµ.

Today

• 1. We will prove Minkowski’s Inequality:

f + gp ≤ f p + gp.

• 2. We will discuss how to make Lp into a normed space called Lp.
• 3. We will prove that the Lp spaces are Banach spaces.
• 4. In particular L2 is a Hilbert space with inner product given by

f , g =

• Ω

f g dµ.

Today

• 1. We will prove Minkowski’s Inequality:

f + gp ≤ f p + gp.

• 2. We will discuss how to make Lp into a normed space called Lp.
• 3. We will prove that the Lp spaces are Banach spaces.
• 4. In particular L2 is a Hilbert space with inner product given by

f , g =

• Ω

f g dµ.