Inner products, operators, equality

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In summary, the conversation discusses whether two operators T and S in Hilbert space are equal if they satisfy the condition (Tf|f)=(Sf|f) for all f in H. It is argued that this condition is not sufficient, as it only implies that (T-S)f is orthogonal to f for all f. However, it is later clarified that if a bounded operator T in a complex inner product space satisfies (x,Tx)=0 for all x, then T must be equal to 0.
  • #1
jostpuur
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Is it true, that if in Hilbert space, operators T and S satisfy (Tf|f)=(Sf|f) for all f in H, then T=S?

I think it is clear, that if (Tf|g)=(Sf|g) is true for all f and g in H, then T=S, but I'm not sure if it is sufficient to only allow g=f.
 
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  • #2
it implies that for all f, (T-S)f is orthogonal to f, but this does not imply T-S is zero. think of a simple example in R^2.
 
  • #3
Ok. T-S can be made a rotation by angle pi/2. :redface:
 
  • #4
I have just learned, that if a bounded operator T in a complex inner product space V satisfies

[tex]
(x,Tx)=0\quad\forall x\in V
[/tex]

then T=0. :wink: I posted the OP after seeing this being used, but didn't understand what was happening.
 
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1. What is an inner product?

An inner product is a mathematical operation that takes two vectors as inputs and produces a scalar value as the output. It is also known as a dot product or scalar product.

2. What is an operator in mathematics?

In mathematics, an operator is a symbol or function that performs a specific operation on one or more inputs to produce an output. Examples of operators include addition, subtraction, multiplication, and division.

3. How do inner products relate to operators?

Inner products and operators are closely related as inner products can be viewed as a type of operator that operates on vectors to produce a scalar value. In fact, many operators in mathematics can be written in terms of inner products.

4. What is the significance of equality in inner products and operators?

Equality is an important concept in inner products and operators because it allows us to determine if two vectors or operators are equivalent. In order for two vectors or operators to be considered equal, they must produce the same output for all possible inputs.

5. How are inner products and operators used in real-world applications?

Inner products and operators are used in a variety of fields, including physics, engineering, and computer science. They are used to model and solve problems related to vectors, motion, and optimization. Examples of real-world applications include calculating work and energy in physics, designing computer algorithms, and analyzing data in machine learning.

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