Hilbert Spaces Flashcards

1
Q

Define an inner product

A
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2
Q

Define pre hilbert space

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3
Q

State Cauchy Schwarz

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4
Q

The inner product on a cartesian product is

A
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5
Q

State continuity in inner products

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6
Q

Define Hilbert space

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7
Q

Define orthongonality

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8
Q

State Pythagorean Theorem

A

If all orthogonal

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9
Q

Is the orthogonal complement closed?

A
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10
Q

State the projection theorem

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11
Q

Define the direct sum

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12
Q

If M is a closed subset of H how can we write H

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13
Q

Define unordered sums

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14
Q

When does an unordered sum converge

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15
Q

State Bessel’s inequality

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16
Q

When a sum of numbers is finite what does that say about the indexing set

17
Q

Given an orthonormal set in a Hilbert space TFAE

18
Q

Define complete orthonormal subset

19
Q

Define maximal orthonormal subset

20
Q

Define partial order

21
Q

Define totally ordered

22
Q

Define upper bound and maximal

23
Q

State Zorn’s Lemma

24
Q

Does every Hilbert space have a basis?

25
Q

What is the relationship between seperable and a basis

26
Q

State the Reisz Representation Theorem for Hilbert spaces