Question 2 Let V be an inner product space with inner norm ||| = VI·I. Prove the parallelogram law ||I+ y||² + ||x – y|l = 2 (||7||² + ||y|²) - This can be interpreted as saying that the sum of the squares of the lengths of the four sides of a parallelogram equals the sum of the squares of the di- agonals.
Addition Rule of Probability
It simply refers to the likelihood of an event taking place whenever the occurrence of an event is uncertain. The probability of a single event can be calculated by dividing the number of successful trials of that event by the total number of trials.
Expected Value
When a large number of trials are performed for any random variable ‘X’, the predicted result is most likely the mean of all the outcomes for the random variable and it is known as expected value also known as expectation. The expected value, also known as the expectation, is denoted by: E(X).
Probability Distributions
Understanding probability is necessary to know the probability distributions. In statistics, probability is how the uncertainty of an event is measured. This event can be anything. The most common examples include tossing a coin, rolling a die, or choosing a card. Each of these events has multiple possibilities. Every such possibility is measured with the help of probability. To be more precise, the probability is used for calculating the occurrence of events that may or may not happen. Probability does not give sure results. Unless the probability of any event is 1, the different outcomes may or may not happen in real life, regardless of how less or how more their probability is.
Basic Probability
The simple definition of probability it is a chance of the occurrence of an event. It is defined in numerical form and the probability value is between 0 to 1. The probability value 0 indicates that there is no chance of that event occurring and the probability value 1 indicates that the event will occur. Sum of the probability value must be 1. The probability value is never a negative number. If it happens, then recheck the calculation.
![### The Parallelogram Law in Inner Product Spaces
**Question 2**
Let \( V \) be an inner product space with an inner norm \(\| x \| = \sqrt{x \cdot x}\). Prove the parallelogram law.
The parallelogram law states:
\[
\| x + y \|^2 + \| x - y \|^2 = 2 \left( \| x \|^2 + \| y \|^2 \right).
\]
This can be interpreted as saying that the sum of the squares of the lengths of the four sides of a parallelogram equals the sum of the squares of the diagonals.
### Explanation
In the above formula:
- \(\| x \|\) represents the norm or the length/magnitude of vector \( x \) in the inner product space.
- \(\| x + y \|\) denotes the norm of the vector resulting from the addition of vectors \( x \) and \( y \).
- \(\| x - y \|\) denotes the norm of the vector resulting from the subtraction of vector \( y \) from vector \( x \).
The parallelogram law demonstrates a fundamental property of inner product spaces, relating to how lengths (or norms) behave under addition and subtraction of vectors. This law is foundational in linear algebra and vector space theory.
### Intuition Behind the Parallelogram Law
Geometrically, imagine a parallelogram:
- The vectors \( x \) and \( y \) can be thought of as the sides of the parallelogram.
- The vectors \( x + y \) and \( x - y \) represent the diagonals of the parallelogram.
The parallelogram law then tells us that the combined length of the diagonals, measured as the sum of the squares of their lengths, is equal to twice the sum of the squares of the lengths of the sides. This corresponds to:
- Summing the distances along the geometry's diagonals.
- Reflecting the inherent symmetry and properties preserved in the structure of inner product spaces.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9a4dcd82-baf4-45bc-b40b-693a3e683492%2F1bf8fcdb-d612-43b7-b573-8aeb2979f404%2F8hwx2jn_processed.jpeg&w=3840&q=75)
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