View search by multiple values, how?
If you are trying to search through a view's node title field, node body field, or some other string field specifically, you may be able to customize the filter how you would like:


Add filter Node: Title or Node: Body
Set the Operator to Contains any word
Click Expose
Click Update


This will display a filter to the user that can filter the view by the corresponding field values that contain any of the entered words, separated by spaces. This does not format the textfield into a textarea, but works functionally as you had asked.