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 orthogonal projection operators. 3. We introduced the L p spaces: � � � f is A -measurable L p = � f : Ω → C , p ∈ (0 , ∞ ) . Ω | f | p d µ < ∞ � � � 4. And their natural seminorm � 1 �� | f | p d µ p � f � p = . Ω 5. We proved that the L p spaces are linear spaces. 6. We proved H¨ older’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 orthogonal projection operators. 3. We introduced the L p spaces: � � � f is A -measurable L p = � f : Ω → C , p ∈ (0 , ∞ ) . Ω | f | p d µ < ∞ � � � 4. And their natural seminorm � 1 �� | f | p d µ p � f � p = . Ω 5. We proved that the L p spaces are linear spaces. 6. We proved H¨ older’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 orthogonal projection operators. 3. We introduced the L p spaces: � � � f is A -measurable L p = � f : Ω → C , p ∈ (0 , ∞ ) . Ω | f | p d µ < ∞ � � � 4. And their natural seminorm � 1 �� | f | p d µ p � f � p = . Ω 5. We proved that the L p spaces are linear spaces. 6. We proved H¨ older’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 orthogonal projection operators. 3. We introduced the L p spaces: � � � f is A -measurable L p = � f : Ω → C , p ∈ (0 , ∞ ) . Ω | f | p d µ < ∞ � � � 4. And their natural seminorm � 1 �� | f | p d µ p � f � p = . Ω 5. We proved that the L p spaces are linear spaces. 6. We proved H¨ older’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 orthogonal projection operators. 3. We introduced the L p spaces: � � � f is A -measurable L p = � f : Ω → C , p ∈ (0 , ∞ ) . Ω | f | p d µ < ∞ � � � 4. And their natural seminorm � 1 �� | f | p d µ p � f � p = . Ω 5. We proved that the L p spaces are linear spaces. 6. We proved H¨ older’s Inequality.
Today 1. We will prove Minkowski’s Inequality: � f + g � p ≤ � f � p + � g � p . 2. We will discuss how to make L p into a normed space called L p . 3. We will prove that the L p spaces are Banach spaces. 4. In particular L 2 is a Hilbert space with inner product given by � � f , g � = f g d µ. Ω
Today 1. We will prove Minkowski’s Inequality: � f + g � p ≤ � f � p + � g � p . 2. We will discuss how to make L p into a normed space called L p . 3. We will prove that the L p spaces are Banach spaces. 4. In particular L 2 is a Hilbert space with inner product given by � � f , g � = f g d µ. Ω
Today 1. We will prove Minkowski’s Inequality: � f + g � p ≤ � f � p + � g � p . 2. We will discuss how to make L p into a normed space called L p . 3. We will prove that the L p spaces are Banach spaces. 4. In particular L 2 is a Hilbert space with inner product given by � � f , g � = f g d µ. Ω
Today 1. We will prove Minkowski’s Inequality: � f + g � p ≤ � f � p + � g � p . 2. We will discuss how to make L p into a normed space called L p . 3. We will prove that the L p spaces are Banach spaces. 4. In particular L 2 is a Hilbert space with inner product given by � � f , g � = f g d µ. Ω
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